Measure Theory 8 | Monotone Convergence Theorem (Proof and Application)

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This is part 8 of 22 videos.

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I hope that this helps students, pupils and others. Have fun!

(This explanation fits to lectures for students in their first and second year of study: Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on)

[renkousami]

I found a bit hard to follow the measure theory course in school and your videos are giving me a great sketch. ty very much^ ^!

[robelbelay]

You have an excellent channel, thank you so much for your efforts!! I will try to watch all your videos as many times as necessary to understand them deeply. :D

[quantitativeease]

9:46 This didn't confuse me but you might have explicitly written out something about the supremum of h.

[MrWater]

I'm struggling with minute 6.48 and below where union Xn complement has measure zero. You define Xn where fn is measurable, h is measurable (because all step functions are measurable) so their composition is measurable, so Xn must be a measurable set. Then you said that "almost every x in X lies in at least one of the Xn because fn converges to f u-a.e" so their complement must have measure zero by definition of convergence almost everywhere, ok. BUT I didn't get, yet, this part "almost every x in X lies in at least one of the Xn because fn converges to f u-a.e" looks like I missed something important there. Is it because as fn converges u-a.e to f, the in the limit fn=f so fn-f = 0 so as h<=f it must be that fn - (1-eps) >= 0 (in the limit). So there are finitely points that does not accomplish that and infinitely many points that accomplish that?

[coralBlue]

Thank you for posting these videos! At 5:00 shouldn't we say that h <= g \mu a.e.? Otherwise we cannot claim that X tilde complement has measure 0.

[chunxiaozhou]

Very clear explanation! Thank you so much! By the way, is there any similar conclusion for decreasing sequence of measurable functions?

[mrahman]

Hi, I am really enjoying this series on measure theory. Thanks for your videos! I had some trouble grasping the application of monotone convergence theorem to the partial sums. To apply the theorem, we need to know what the partial sums converge to. Could you kindly provide some more details of the application?

[StratosFair]

Why is the measure of the complement of Xtilde 0 ? I know that the sequence fn converges to f, so is the argument that for any epsilon, there is always an integer N for which if n>N, then fn is epsilon close to f ?
If it is the reason, then I don't see why the measure remains always 0, because the complementary of Xtilde gets bigger and bigger as epsilon goes towards 0. Am I missing something ?

[RealAXork]

I'm confused as to what the point of epsilon is in this proof.
You say it's used to adjust the height of h but h is already arbitrary so it will include all possible choices of epsilon as well.
Unless I'm missing something.

Incidentally, couldn't this be proved by considering the sequence f - fn since all of its elements also belong to the familiy f+ and it converges to 0, so the sequence of integrals converge to or are at least bounded by zero so using the linearity of the integral we can say that limit(integral(f - fn)) <= 0 implies limit(integral(f) - integral(fn)) <= 0 which due to f not depending on n implies integral(f) <= limit(integral(fn))
Of course the other inequality is still trivial to prove.

[DanielKRui]

Hi! Thank you so much for your videos --- they are very helpful and well paced (and beautiful too). A little question, what do you mean we have to fill in details on one of the final steps (you wrote it in green ink)? What explicitly do we need to show there?

[ocramod]

Thank you very much for the lecture! However, I think the application is missing an unproven corollary (unless I missed something in the previous videos). In particular let $S_k = \sum_{n=1}^k g_n$ being the partial sum of the series. The unproven corollary is the following $\int_X S_k d\mu = \sum_{n = 1}^k\int_X g_n d\mu$. This is not trivial (at least not to me that I am not a mathematician). To prove that, one has to prove first that $\int_X (f+g) d\mu = \int_X fd\mu+\int_X g d\mu $ (which involves of course the MCT but not only that). After we prove this then we can easily prove by induction the ``missing corollary." Once we have this then I agree that the application becomes a straightforward application of the MCT.

[Ryanxyang]

It would be helpful if you briefly explain why epsilon is needed. Without it, Xn may not converge to X, if h overlaps with f over a set with non-zero measure.

[samtux]

Does the monotone convergence apply to the classical integral (as defined by Newton) or it only applies to the Lebegue integral?

[algorithmo]

is there a video on convergence of fourier series, stone weirstrass theorem?

[ibrahimislam]

Can you please just clear my confusion;we studies two very important theorems and those are Dominated convegence theorem and monotone convergence theorem, my question is:why is this property "limit of integral is equal to integral of limit" is so important? I mean I studies in textbooks that the major benefit of using lebesgue integration over Riemann integration is that we can push limit inside integral this is natural question which comes to mind that why this property is so important?could you please explain?

[farbodtorabi]

Could you please clarify how you went from limits to sums? I'm a bit confused how the Application follows from the Proof. In the Proof you're talking about limits being interchangeable with the integral, but in the application you interchanged the infinite sum with the integral, how is that possible? Thank you :)

[HuangDuang]

I have a question about the application example at the end of this video, why it need't assume gn is monotony? I mean the name of the theory is called monotone convergence theorem

[alextremayne]

How is that an application of the MCT?
It's not clear at all, am I just missing something by only watching and not note-taking?

[jacobhelwig]

So the "details for you" at 8:18 was showing that as n goes to infinity, the integral over X_n of (1-epsilon)h goes to the integral over the union of X_n of (1-epsilon)h...is the reason that this is true because this is a limit of a monotonic sequence of sets (that is, X_{n-1} is a subset of X_n) which converges to the union? Or is this not generally true and only if the integrand is a simple function?

[nathaliebastoaguirre]

Love your videos, do you have some for convex analysis? or convex optimization?. thank you for all the videos