Let fn be a sequence of measurable functions. Show that the set E= {x: fn(x). converges } is mea

step 1 Okay, good day, ladies and gentlemen. Today we're going to look at a problem that is a proof, ba

Let fn be a sequence of measurable functions. Show that the...

Key Concepts

Limit Superior and Limit Inferior

The limit superior (limsup) and limit inferior (liminf) provide ways to characterize the convergence of a sequence of functions by capturing the eventual upper and lower bounds of the sequence at each point. The equality of the limsup and liminf characterizes the points where the sequence converges and these constructions are measurable when applied to measurable functions.

Measurable Functions

A measurable function is one for which the preimage of any Borel set is measurable. This concept is pivotal in measure theory as it allows integration and the manipulation of functions in a measurable space, and it underpins many results including limit theorems.

Pointwise Convergence

Pointwise convergence of a sequence of functions refers to the property that for each individual point in the domain, the sequence converges to some limit. Studying pointwise convergence involves investigating the set of points where the sequence converges, which is important for understanding the behavior of sequences of measurable functions.

Preservation of Measurability Under Countable Operations

Measurable sets are closed under countable unions, intersections, and other operations. Since operations like taking limsup, liminf, and equality are formed via countable processes, this property ensures that the set of points at which a sequence converges remains measurable.

Transcript

00:01 Okay, good day, ladies and gentlemen.

00:05 Today we're going to look at a problem that is a proof, basically, which can ask us to prove.

00:13 You know, we're assuming that is positive, continuous and decreasing, and that we have this property here, this property here.

00:25 And we know that this converges to us, that we want to show that our end.

00:33 Here is less than i guess this is this integral here.

00:39 Okay, so first point i want to make here is that since the series converges we know from one of our, from one of the earlier results that this integral actually exists.

00:57 And this is this is pretty important to do anything really you need to know that the integral itself exist, which of course means then that this integral also exists.

01:14 So we can talk about this guy unambiguously and everything we do.

01:21 So that's pretty important.

01:23 That's pretty important in the whole thing.

01:26 So next, i want to point out something that i've talked about earlier and i'll just point it out again.

01:37 Now, what we're doing here is really what we've called before lower remand sums.

01:47 So this is lower remand sums, okay? and i think by now you should have done these for other cases.

02:02 The idea is that if you have an integral, you can approximate it from both above and below.

02:15 So if you know the integral exists, you can approximate it from above and below by what we call remand.

02:22 Sums and the basic idea is you start with now lower remon sums basically we set out some points in this case i have n minus one and and plus one and plus two and you draw these rectangles here now the difference here or the the length here is is what we sometimes refer to as delta act in this case, it's equal to one.

02:56 And the height here is, of course, equal to f of n.

03:03 The lower remand sums mean we use this point here, this endpoint...

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