Measure Theory

Here, you find my whole video series about Measure Theory in the correct order and I also help you with some text around the videos. If you have any questions, you can use the comments below and ask anything. However, without further ado let’s start:

If you want to learn Measure Theory and understand the general Lebesgue Integral, you first need to know what a Sigma-algebra is:

Now you know that a Sigma-algebra is just a special collection of subsets. Of course, if I give you just any collection of subsets, you are able to check if this one satisfies the three rules above. If this is not the case, you can add new sets into the collection such that the result is a sigma-algebra. In the next video you see how this works in a theoretical way (generated Sigma-algebra).

One other collection of subsets is given in a topology: open sets! With these, we can generate a very important Sigma-algebra: the Borel Sigma-algebra.

You might have already noted that the whole point of the Sigma-algebra is that we can define a map from it to the real numbers. This map should measure the set in a sense of a generalised volume. Therefore, we just call this map a measure:

As an interlude I want to answer the question why we don’t just ignore Sigma-algebras and define the measure on the whole power set. This would mean that we can measure “volumes” of all possible subsets and wouldn’t that be a reasonable request? Yeah, it would be. However, then we would lose a lot of interesting measures, especially the most important one: The usual volume measure in $ \mathbb{R}^3 $. The next video proves that we cannot have this volume measure (and other ones) defined for the whole power set:

Now, we can go deeper into the theory. Keep in mind that someday we want to define a generalised integral with the help of such a measure. For that reason, we need to look at special maps that converse the structure of our $ \sigma $-algebra: measurable maps.

At this point, we can start by defining the so-called Lebesgue integral. It is the modern integral notion that even works in this abstract framework. However, let’s start with simple functions: step functions!

Wow, now we know what $ \int f d \mu $ means. It is a number that really represents the orientated volume for the graph of the function. Everything seems easier to define than what you might know from the Riemann integral. However, the real advantages of the Lebesgue integral come into play when we look at the convergence theorems:

Also a proof would be nice:

Related to the convergence theorems is a very general fact, now often just called Fatou’s Lemma:

At this point, you are so familiar with the Lebesgue integral that the next theorem might not surprise you so much. However, it is a property the simple one-dimensional Riemann integral lacks, but it has many applications. There, Lebesgue’s dominated convergence theorem is might favourite theorem in the integration theory:

Of course, you want to see a proof of this beautiful theorem, so here we go:

I hope that you now have a good overview about measures and the Lebesgue integral. We now go one step back and look at the foundations again. Especially, we are interested how we can construct some particular measures like the normal Lebesgue measure, for example. Therefore, let’s first talk about CarathĂ©odory’s extension theorem:

With the help of this theorem, we are able to construct a lot of different measures. One special class are so-called Lebesgue-Stieltjes measures:

We reached a point where we talk about deeper and special results in Measure Theory. However, both theorem that now follow have so many applications that you could come across them everywhere. They are called the Radon–Nikodym theorem and Lebesgue’s decomposition theorem:

For the Riemann integral, you know how to apply a substitution to solve some integrals. Of course, this also works for the Lebesgue integral in $ \mathbf{R} $ in the same way. More interesting is the fact that you can generalise this rule to all other measurable spaces. There we need to define the image measure to get a similar substitution rule:

Again, I should show you a proof of this:

Now you know how to get new measures when you have a measurable map. It works similar when you want to define a new measure for the cartesian product. This one is called the product measure. With the help of this, we can simply higher-dimensional integrals. This is known as Cavalieri’s principle:

Of course, an example would be helpful:

We came far into the topic of measure theory and integration theory. I want to close the last topic but presenting one of the most famous theorems in this context: Fubini’s Theorem. You might already heard it and even applied it but I want to show you the general context such that you know what the correct assumptions are. I also show you an example 🙂

The next videos are more about the foundations of measure theory. Especially we first talk about outer measures in the next two videos:

For the moment this ends my series about measure theory. Someday, I will continue the series by showing you the construction of the Lebesgue measure.