As a Phd in statistics the hardest part for me about math in grad school was the why (or ideas). Understanding the why will motivate any hardworking student to fill in all the technical details. You make the why so easy! This is what makes mathematics such a fun subject. Even as a hobby. I'd like to see you write a book. Thank you for sharing your knowledge.
I am also taking PhD Statistics. Our prof in Real Analysis rarely explains the "underlying principles" of analysis. I commend this channel for really showing #thebrightsideofmathematics
Exactly! Like if someone told you to run into the forest, the "why" is a big factor in giving you motivation, say a bear at loose. This is what professors many times do not get.
I am an electronics engineer, now pursuing higher studies and I've always loved maths - your videos are amazing! Short with crystal clear explanations following the classic definition-theorem-proof-example style. I rarely subscribe to channels but I am really looking forward to more videos on further topics, so I did...please keep making more!
This video was so awesome. I'm a physicist, computer scientist and data scientists and I always find these concepts but without a formal explanation. Every time I looked for the formal explanation it was cryptic. This is so clear. Thanks a lot!
I'm a mathematics enthusiastic and each time I read something about maths I get in contact with new mathematical 'creatures' like Measure Theory... sometimes they are just rigorous descriptions of things I already knew.
This elucidates the measure theory in a quite clear way. I have been struggling to understand what sigma-algebra or Borel-sigma is for many days. Somehow, your video eliminates my doubts. Excellent work!
Great introduction! Love the balance between intuition and formalism. I was trying to learn this from the last chapter in Rudin 1 but was understanding close to nothing
Hi! When is the Lebesgue Measure video coming out? Could you include an explanation on how the outer measure & visual representation of the infimum of the coverings?
What is a measure? More like "This video's a treasure!" Thanks for another illuminating demonstration about a subject which has seemed impenetrable to me.
15:50 Lebesgue measure extends the two properties of volume and translation to arbitrary subsets. The power set cannot be chosen as the sigma-algebra for the Lebesgue measure, rather a smaller one is chosen, called the Borel sigma-algebra.
Thanks for the great videos! These are so helpful. I do have a question. At 4:40 or so, you say that A_i and A_j are pairwise disjoint, i.e. A_i intersect A_j = {}. But this seems to create ambiguities regarding the decision of where the boundary points of the two sets belong, because if they are included in one subset or the other (but not both), then one of the sets is half-closed while the other is half-open, meaning we cannot subdivide sets into closed subsets or open subsets automatically. If we take a simple 1-dimensional case and consider the closed interval [0,1] cut into two equal-sized pieces, it's unclear whether we use [0, 0.5] and (0.5, 1], or [0, 0.5) and [0.5, 1], but either way, we no longer have closed subintervals of our closed unit interval, which is annoying if we want to consider a large collection of such subintervals while taking advantage of them all being closed. (The same problem arises if we start with an open interval and want to subdivide it using open subintervals--we end up with intervals that are half-open or -closed.) Stein and Shakarchi appear to resolve this issue in their analysis text (Vol. 3) by using the term "almost disjoint" (p. 4), meaning the intersection of any two A_i is empty except for the boundary, so in my scenario they would permit A_1=[0, 0.5] and A_2=[0.5, 1] (i.e., they overlap but only at the boundary point 0.5). Then everything they construct is premised on the notion of unions of nearly disjoint sets. Do you mean nearly disjoint or indeed truly disjoint (and if so can you please explain whether there is some convention for handling boundary points)? Please let me know and thanks again!
@@brightsideofmaths ok thanks so much for responding! I think I answered my own question by realizing this is probably equivalent to Stein. Under your approach, a boundary point must be contained in one subinterval or subregion but not both. For instance, if we're on the closed unit interval and we partition in half, we either get [0, 0.5], (0.5, 1], or [0, 0.5), [0.5, 1]. But for any half-open/half-closed cube E in R_d, we can consider a completely open cube F contained inside such half-open/half-closed cube, such that the measure of the symmetric difference between the two sets is less than epsilon over 2^j. I think we can also do something similar with a closure. Am I correct that these are all equivalent because they differ only by a collection of points that have measure 0? Thanks again!!
You have mentioned 8:00 for all Ai in A(sigma alg) and also that Ai intersection Aj should be null. But in sigma algebra there are some groups of sets that have something in common. eg. full set and any other. I think you should put a different notation on the right side of (b) not using the index i
Hey there! I am a romanian math student and I want to help teenagers with math problems ( mostly college not university, I'm not that smart yet :)) ) and I love watching your videos so I want to do the same but in romanian so that everyone in my country can understand. Can u tell me with what do you edit your videos and what exactly do you use to make your videos?
Around 8:50, how do you get the finite version of rule (b) from the infinite version? And I can't believe you do these in German and English. I think you are doing a great job with the explanations and the use of technology here. I really do hope we'll see more of this style of presentation on the web in the future.
1:39 I've seen abuse of notation before but this needs a trigger warning! I'm loving the videos though. One great thing about math is how you can learn the same concept again from a different perspective, and that enhances appreciation of the topic and opens the mind to its generalizations. 8:32 LOL do you mean that nothing should have a zero volume or nothing should have a zero volume?
Thank you :) The included infinity symbol is useful and I explained it immediately, so this abuse of notation should not be a problem. Yeah, I am also not so happy with 8:32 because "nothing" should be the "empty set" :D
@@brightsideofmaths To be clear, it was the idea of infinity being on the "closed" interval I found upsetting. In context it "makes sense" since it refers to unions of countably infinite and disjoint sets. It is easy to equate the union of sets with arithmetic addition but there are some important differences.
I was going to comment on the zero volume, but then i saw your comment :-) Irrelevant to the video: Can you (or anyone) please explain why do we not care about uniform probability measure on a countably infinite space? I can understand the impossibility result for uncountable set which says that the sigma algebra cannot be the power set if we want to define a uniform probability measure. My question is, why don't we use the same remark for countably infinite set, after all we can't assign same nonzero probability measure to all singletons in the set of natural numbers.
@@sunilrampuria9339 I'm sorry, but I do not know what a "uniform probability measure" is. If you could define it, maybe somebody who watches these videos could give you some insight.
I have a question regarding the σ-additivity of a measure μ. Since we are working with infinite sums, how do we guarantee convergence? Is there some condition that is missing that is guaranteeing convergence?
That is a very good question. However, the answer is simple: In the infinite sum, there are only non-negative numbers. So either the sum is convergent or it is not but then we just get the symbol ∞ as an outcome.
Hi, Thanks for a great video. However I have a question regarding the counting measure example. Isn't the 2nd rule violated in that case? The union of two sets {a,b} and {a,c} will give you a counting measure = 3 for mu( {a,b,c} ) However on the right hand side of the equation you would get 4? because you add mu(a,b) + mu(b,c) and get the answer 4?
Great video. I am a bit confused about why you always keep saying volume and not length since you are in one dimension. Everything you are defining is done in intervals of real line and those intervals can only have length not volume. What am i missing?
Thank you. We have an abstract measure space X. This could be, for example, R^3 and then "volume" makes sense. Don't mix the sides: We measure the volume with a single real number but the object, we measure, would still live in R^3.
Just to clarify: a map is a measure if it satisfies (a) and (b), but (b) can imply (a), right? So that a map is a measure if it satisfies (b) should be fine, no?
The Bright Side Of Mathematics yes, never stop. You are helping us learn measure theory. You are a true professor with a gift of teaching. May you always teach and I hope you are blessed with a long, blissful life!
How does the Dirac measure deal with the second rule for measure? If A1 and A2 both contain the point, then u(A1) = 1 and u(A2) = 1, so u(A1 U A2) = sum (u(A1), u(A2) = 2. However, because of union rule for measurable sets, A1 U A2 is a measurable set, which we call A3, then its measure u(A3) = 1. This is a contradiction.
Ich weiß jetzt nicht wie dein Vortrag zur Maßtheorie weiter verläuft, aber ich hätte zumindest den Unterschied zwischen einem Prämaß und Maß dargestellt. Es war zumindest in den Vorlesungen wichtig.
Hi thank u for ur great efforts, but a question, y u don't use numbered examples (examples that contain numbers and not letters) to explain the definitions, axioms, or any explanations?
@@brightsideofmaths Thank you for your feedback. Unfortunately I couldn't continue the whole series because it was little bit hard to me to catch mathematics issues without numbers, dass heiß, this was the last lecture I saw, so i don't know if the further lessons have numbered examples or no. If they don't, I strongly recommend to write an example as audience can read it with of course numbers and then solve this numbered example. If find easier and even faster to understand anything in math. Anyway, good luck, what you are doing auf jeden Fall very great.
I'm probably late to this party, but those look like Kolmogorov axioms. Supposing a normed measure (I dunno what to call it) is a measure of codomain [0,1], probabilities are like volumes of event-space, right?
This party is still going on ;) For probability measures, you can look here: th-cam.com/video/u5IouBwYji4/w-d-xo.html Probability measures are just normed measures or normed volumes.
He says in the video that he will show that the Borel sigma algebras are able to construct such a proper measure function. Does anyone know in which video he shows that?
Sigma algebra is enough for measure, why there is borel subset? Borel set is created from open sets, but it includes closed sets according to the definition of sigma algebra, since complementary of an open set is closed set.
It is a matter of speaking. Just try to collect all instances where one uses "volume" in everyday life: File storage, speaker loudness and so on. It is the perfect word to make sense as a n-dimensional volume here.
It is hard for me to visualize the sigma algebra generated by, say, the natural or the real numbers. I'm trying to visualize what these sets look like. And also what sets from the power set are absent from the sigma algebra. Is there an intuition to understand which sets are measurable in such cases?
@@brightsideofmaths Thanks! It makes excellent sense. I just remembered that the places I saw Dirac mass and Dirac measure used interchangeably are in optimal transport literature.
Hum... I think that we need a little more to have raisonnable measure on R^n extending classical volume definition. In particular, not only invariance under translation but also with respect to all rigid motions (what about rotations ?). Maybe is it possible to construct a measure, finite for the unit cube, invariant by translation but not invariant under rotations. Very nice video... Edit : Due to the unicity of the Carathéodory's extension theorem, it is not possible to construct another measure than the Lebesgue measure that is invariant under translations only (and equal to one on the unit cube). At least on the Borel sets.
At 7:50, when defining what a measure is, don't we also need to include that the measure applied to an element of the sigma algebra should be greater than or equal to zero? i.e. µ(A) ≥ 0 for every A ∈ F , where (X, F) is the measure space
Of course, we need this property. I have hidden this in the description of the map µ itself: It has to be a map from the sigma-algebra into the interval [0,inf].
I really dont know what is the difference between the indicator function and dirac measure. I mean one is a measure and other is a function but they behave EXACTLY the same.
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As a Phd in statistics the hardest part for me about math in grad school was the why (or ideas). Understanding the why will motivate any hardworking student to fill in all the technical details. You make the why so easy! This is what makes mathematics such a fun subject. Even as a hobby. I'd like to see you write a book. Thank you for sharing your knowledge.
agree!! I m so frustrated why no professors ever talked about why we study those thing how did they come up?
I am also taking PhD Statistics. Our prof in Real Analysis rarely explains the "underlying principles" of analysis. I commend this channel for really showing #thebrightsideofmathematics
Indeed. As a Phd in computer science, I found this tutorial is so helpful for me to understand the question I never asked before - "Why".
seems that grad student are learning their trade on TH-cam. so, why do they go to grad school in the first place? I know. It is the cert that matters.
Exactly! Like if someone told you to run into the forest, the "why" is a big factor in giving you motivation, say a bear at loose. This is what professors many times do not get.
I am an electronics engineer, now pursuing higher studies and I've always loved maths - your videos are amazing! Short with crystal clear explanations following the classic definition-theorem-proof-example style. I rarely subscribe to channels but I am really looking forward to more videos on further topics, so I did...please keep making more!
Thank you very much :) More videos are coming!
@@brightsideofmaths
Sex
This video was so awesome. I'm a physicist, computer scientist and data scientists and I always find these concepts but without a formal explanation. Every time I looked for the formal explanation it was cryptic. This is so clear. Thanks a lot!
Thanks a lot! And thanks for the support :)
God bless you Mr Bright side. You are really making Mathematics BRIGHT. Keep it on
very helpful in understanding. i request you make videos in very topic of mathematics.
I appreciate these videos on measure theory and bless you for adding in examples with boxes because that clarified things so much.
Thank you for those videos, they really help me lay a foundation for more formal probability theory!
Thank you so much for your lucid explanations!
Your videos make me love mathematics
You sir have a real talent for teaching
Thanku so much sir .you are a perfect teacher. Hats off you for your presentation.
I'm a mathematics enthusiastic and each time I read something about maths I get in contact with new mathematical 'creatures' like Measure Theory... sometimes they are just rigorous descriptions of things I already knew.
This elucidates the measure theory in a quite clear way. I have been struggling to understand what sigma-algebra or Borel-sigma is for many days. Somehow, your video eliminates my doubts. Excellent work!
I really like how you do the curly brackets!
The content is also not bad, keep it up!
Amazing video! Amazing series! Please keep it coming! Measure theory has never been easier to understand. Thank you!!
Great introduction! Love the balance between intuition and formalism. I was trying to learn this from the last chapter in Rudin 1 but was understanding close to nothing
Glad it was helpful! :)
Hi! When is the Lebesgue Measure video coming out? Could you include an explanation on how the outer measure & visual representation of the infimum of the coverings?
The next videos are coming out in 1-2 weeks :)
You're increidble in making the complicated seem so easy you were a great help for me in revising for my exams please keep sharing your knowledge
Thank you so much :)
How did your exams go?
Excellent series. Looking forward to part 4.
I really enjoy your lectures.
Many thanks!
What is a measure? More like "This video's a treasure!" Thanks for another illuminating demonstration about a subject which has seemed impenetrable to me.
Let Ω be anonempty set and A⊆ B ⊆P(Ω).
Then
ℴ ⊆ ℴ
شكر لا متناهي لكم.
merciiii infiniment
..a wait for part 3 is just finally over..
Thank you very much indeed😊😊
15:50 Lebesgue measure extends the two properties of volume and translation to arbitrary subsets. The power set cannot be chosen as the sigma-algebra for the Lebesgue measure, rather a smaller one is chosen, called the Borel sigma-algebra.
you are great. your lectures is very interesting
Thanks a lot :)
Great lesson, easy to understand Thanks!!
Excellent lecture, thank you sir.
Thank you for showing Dirac measure and the count measure. In the same sense can we say dot product as a measure on a suitable measure space?
Thank you very much? What is your definition of the dot product?
thank you so much very helpful videos
I finally understand "translation invariant".
Thanks for the great videos! These are so helpful.
I do have a question. At 4:40 or so, you say that A_i and A_j are pairwise disjoint, i.e. A_i intersect A_j = {}. But this seems to create ambiguities regarding the decision of where the boundary points of the two sets belong, because if they are included in one subset or the other (but not both), then one of the sets is half-closed while the other is half-open, meaning we cannot subdivide sets into closed subsets or open subsets automatically. If we take a simple 1-dimensional case and consider the closed interval [0,1] cut into two equal-sized pieces, it's unclear whether we use [0, 0.5] and (0.5, 1], or [0, 0.5) and [0.5, 1], but either way, we no longer have closed subintervals of our closed unit interval, which is annoying if we want to consider a large collection of such subintervals while taking advantage of them all being closed. (The same problem arises if we start with an open interval and want to subdivide it using open subintervals--we end up with intervals that are half-open or -closed.) Stein and Shakarchi appear to resolve this issue in their analysis text (Vol. 3) by using the term "almost disjoint" (p. 4), meaning the intersection of any two A_i is empty except for the boundary, so in my scenario they would permit A_1=[0, 0.5] and A_2=[0.5, 1] (i.e., they overlap but only at the boundary point 0.5). Then everything they construct is premised on the notion of unions of nearly disjoint sets. Do you mean nearly disjoint or indeed truly disjoint (and if so can you please explain whether there is some convention for handling boundary points)? Please let me know and thanks again!
Thanks for the comment! We mean disjoint as in the definition: the intersection is empty.
@@brightsideofmaths ok thanks so much for responding! I think I answered my own question by realizing this is probably equivalent to Stein. Under your approach, a boundary point must be contained in one subinterval or subregion but not both. For instance, if we're on the closed unit interval and we partition in half, we either get [0, 0.5], (0.5, 1], or [0, 0.5), [0.5, 1]. But for any half-open/half-closed cube E in R_d, we can consider a completely open cube F contained inside such half-open/half-closed cube, such that the measure of the symmetric difference between the two sets is less than epsilon over 2^j. I think we can also do something similar with a closure. Am I correct that these are all equivalent because they differ only by a collection of points that have measure 0? Thanks again!!
You have mentioned 8:00 for all Ai in A(sigma alg) and also that Ai intersection Aj should be null. But in sigma algebra there are some groups of sets that have something in common. eg. full set and any other.
I think you should put a different notation on the right side of (b) not using the index i
Thank you
You're welcome
Hey there! I am a romanian math student and I want to help teenagers with math problems ( mostly college not university, I'm not that smart yet :)) ) and I love watching your videos so I want to do the same but in romanian so that everyone in my country can understand. Can u tell me with what do you edit your videos and what exactly do you use to make your videos?
Around 8:50, how do you get the finite version of rule (b) from the infinite version? And I can't believe you do these in German and English. I think you are doing a great job with the explanations and the use of technology here. I really do hope we'll see more of this style of presentation on the web in the future.
Thank you very very much! Great video and didactic!
1:39 I've seen abuse of notation before but this needs a trigger warning! I'm loving the videos though. One great thing about math is how you can learn the same concept again from a different perspective, and that enhances appreciation of the topic and opens the mind to its generalizations.
8:32 LOL do you mean that nothing should have a zero volume or nothing should have a zero volume?
Thank you :)
The included infinity symbol is useful and I explained it immediately, so this abuse of notation should not be a problem. Yeah, I am also not so happy with 8:32 because "nothing" should be the "empty set" :D
@@brightsideofmaths To be clear, it was the idea of infinity being on the "closed" interval I found upsetting. In context it "makes sense" since it refers to unions of countably infinite and disjoint sets. It is easy to equate the union of sets with arithmetic addition but there are some important differences.
@@quantitativeease It is just a one-point compactification. Hence, I see this as a natural notation.
I was going to comment on the zero volume, but then i saw your comment :-)
Irrelevant to the video: Can you (or anyone) please explain why do we not care about uniform probability measure on a countably infinite space? I can understand the impossibility result for uncountable set which says that the sigma algebra cannot be the power set if we want to define a uniform probability measure. My question is, why don't we use the same remark for countably infinite set, after all we can't assign same nonzero probability measure to all singletons in the set of natural numbers.
@@sunilrampuria9339 I'm sorry, but I do not know what a "uniform probability measure" is. If you could define it, maybe somebody who watches these videos could give you some insight.
Clear and inspiring curiosity, thank you very much for your work-
So fantastic and helpful
Nice video presentation.
I have a question regarding the σ-additivity of a measure μ. Since we are working with infinite sums, how do we guarantee convergence? Is there some condition that is missing that is guaranteeing convergence?
That is a very good question. However, the answer is simple: In the infinite sum, there are only non-negative numbers. So either the sum is convergent or it is not but then we just get the symbol ∞ as an outcome.
Hi, Thanks for a great video. However I have a question regarding the counting measure example.
Isn't the 2nd rule violated in that case? The union of two sets {a,b} and {a,c} will give you a counting measure = 3 for mu( {a,b,c} )
However on the right hand side of the equation you would get 4? because you add mu(a,b) + mu(b,c) and get the answer 4?
You are welcome! Don't forget that we need *disjoint* unions for the second property.
Perfekt, Thanks!
thankyou! amazing explanation.
Would it be correct to say, that a measure space is basically the size, or the volume of the particular sigma algebra?
Thank you for your support! A measure space is the collection of set, sigma-algebra and measure.
Great video. I am a bit confused about why you always keep saying volume and not length since you are in one dimension. Everything you are defining is done in intervals of real line and those intervals can only have length not volume. What am i missing?
Thank you. We have an abstract measure space X. This could be, for example, R^3 and then "volume" makes sense. Don't mix the sides: We measure the volume with a single real number but the object, we measure, would still live in R^3.
Nice.Your videos are so helpful.
Thanks a lot for this series :)
My pleasure!
Your videos are so good! Supporting!!!
Just to clarify: a map is a measure if it satisfies (a) and (b), but (b) can imply (a), right? So that a map is a measure if it satisfies (b) should be fine, no?
Great video, thanks!
Can you please do a video on haar measure?
OMG please upload part 4 5 6 7 8 and so on. I am literally begging you!!
I am working on part 4 at the moment. :)
The Bright Side Of Mathematics I appreciate it!!
The Bright Side Of Mathematics yes, never stop. You are helping us learn measure theory. You are a true professor with a gift of teaching. May you always teach and I hope you are blessed with a long, blissful life!
respected sir great work
14:25 example of measure: properties of a Lebesgue measure
How does the Dirac measure deal with the second rule for measure? If A1 and A2 both contain the point, then u(A1) = 1 and u(A2) = 1, so u(A1 U A2) = sum (u(A1), u(A2) = 2. However, because of union rule for measurable sets, A1 U A2 is a measurable set, which we call A3, then its measure u(A3) = 1. This is a contradiction.
Why did you ignore the disjoint condition in the second rule?
Well done!
Me onto your great work again,,,so obsessed 💯
Happy to hear that!
How does the additivity rule work for "Direct measure for a fixed point in X)?
Please upload part 4
you are great, thank you so much!
Thank you man, you are the best
Can a measure be infinity?
Or is that just said to be unmeasurable?
Infinity as a value is totally allowed.
life saver...❤🙏🙏🙏
Thank you :)
@6:50 what have we changed so that we can go from finite number of unions to infinite number of unions?
The point of view. We want to approximate volumes as well.
amazing channel
Thank you, you are the best
i watch ur series and friends think im smart :)
You are amazing. That's it. You are amazing.
Thanks :)
You are so good!
Ich weiß jetzt nicht wie dein Vortrag zur Maßtheorie weiter verläuft, aber ich hätte zumindest den Unterschied zwischen einem Prämaß und Maß dargestellt. Es war zumindest in den Vorlesungen wichtig.
See here: tbsom.de/s/mt
Hi
thank u for ur great efforts, but a question, y u don't use numbered examples (examples that contain numbers and not letters) to explain the definitions, axioms, or any explanations?
Thanks! You mean a complete list for the whole series?
@@brightsideofmaths
Thank you for your feedback.
Unfortunately I couldn't continue the whole series because it was little bit hard to me to catch mathematics issues without numbers, dass heiß, this was the last lecture I saw, so i don't know if the further lessons have numbered examples or no. If they don't, I strongly recommend to write an example as audience can read it with of course numbers and then solve this numbered example. If find easier and even faster to understand anything in math.
Anyway, good luck, what you are doing auf jeden Fall very great.
I'm probably late to this party, but those look like Kolmogorov axioms. Supposing a normed measure (I dunno what to call it) is a measure of codomain [0,1], probabilities are like volumes of event-space, right?
This party is still going on ;) For probability measures, you can look here: th-cam.com/video/u5IouBwYji4/w-d-xo.html
Probability measures are just normed measures or normed volumes.
@@brightsideofmaths Ayy cheers! Also I just want to say your linear algebra series carried me through Lin 2!
He says in the video that he will show that the Borel sigma algebras are able to construct such a proper measure function. Does anyone know in which video he shows that?
The last ones in the series :)
At 0:50 did you mean a sigma algebra is a special collection of subsets of the power set of X?
A collection of subsets of X. Subsets of X are ELEMENTS of the power set of X :)
First, this is complicated but after some time you get used to it :)
@@brightsideofmaths but a sigma algebra on X can be the power set of X itself right? And the power set of X is not a subset of X
@@samernoureddine However, the power set of X is a collection of subsets of X.
@@brightsideofmaths that makes sense now. Thanks! Fantastic videos by the way. Excited to see more of your work
Sigma algebra is enough for measure, why there is borel subset? Borel set is created from open sets, but it includes closed sets according to the definition of sigma algebra, since complementary of an open set is closed set.
The Borel sets are used to form a sigma algebra :)
Why do we say that we are measuring "Volume" why not area, why not distance? Why specifically volume 3 dimensions.
It is a matter of speaking. Just try to collect all instances where one uses "volume" in everyday life: File storage, speaker loudness and so on. It is the perfect word to make sense as a n-dimensional volume here.
👍 You are good soul
Shouldn't the map be (X, A) -> [0; inf] rather than A -> [0; inf] ?
Reading it as a map it's literally A -> [0; inf].
It is hard for me to visualize the sigma algebra generated by, say, the natural or the real numbers. I'm trying to visualize what these sets look like. And also what sets from the power set are absent from the sigma algebra. Is there an intuition to understand which sets are measurable in such cases?
Oh this is the topic of the next video, so I'll look at this before.
In some books, I saw the notion of Dirac mass, is Dirac measure == Dirac mass?
I would not use the term "mass" for a measure but I also have seen it. Maybe it's a bad translation from the German word "Maß".
@@brightsideofmaths Thanks! It makes excellent sense. I just remembered that the places I saw Dirac mass and Dirac measure used interchangeably are in optimal transport literature.
Why is it necessary that the A_i’s are pairwise disjoint? Or perhpas why cannot they coincide?
The sigma-additivity tells you how we can add disjoint volumes. If there was an overlap, the added volume would exceed the volume of the union.
Hum... I think that we need a little more to have raisonnable measure on R^n extending classical volume definition.
In particular, not only invariance under translation but also with respect to all rigid motions (what about rotations ?).
Maybe is it possible to construct a measure, finite for the unit cube, invariant by translation but not invariant under rotations.
Very nice video...
Edit : Due to the unicity of the Carathéodory's extension theorem, it is not possible to construct another measure than the Lebesgue measure that is invariant under translations only (and equal to one on the unit cube). At least on the Borel sets.
Ooooooooo nice!!
Thanks :)
At 7:50, when defining what a measure is, don't we also need to include that the measure applied to an element of the sigma algebra should be greater than or equal to zero?
i.e. µ(A) ≥ 0 for every A ∈ F ,
where (X, F) is the measure space
Of course, we need this property. I have hidden this in the description of the map µ itself: It has to be a map from the sigma-algebra into the interval [0,inf].
@@brightsideofmaths ah yes, thanks for the clarification
I really dont know what is the difference between the indicator function and dirac measure. I mean one is a measure and other is a function but they behave EXACTLY the same.
What do you mean by "behave"?
As in the usage and result. Both have a set input both give 1 for a single value...@@brightsideofmaths
But the inputs are different :)
100.000 klicks mit Maßtheorie, wow :). Verdient!
Thanks!
any real life application?
Sure! Many :)
Hah, this makes what i read make a lot more sense. Seesh, I wish I was smart enough to derive my own examples.
Don't we also need to discuss outer measures?
That depends what you want to prove. In the end of this series, I will add some proofs and also discuss outer measures.
I'm annoyed that you use two strokes to make brackets, but they look nice enough for me to be fine with it
Okay, I try to change that :)