## Integrable functions

We’ve finally reached the last part of the construction of the lebesgue integral! So far we have successfully defined the integral on non-negative measurable functions. However, we now want to generalize the defintion to function that can be negative. How would you do it? If you were told by your professor to find a defintionContinue reading “Integrable functions”

## Almost everywhere – the new everywhere

After defining the lebesgue integral for non-negative measurable functions, proving the monotone convergence theorem, fatou’s lemma and lots of properties of the integral – we are finally ready to move on to the last step of the definition. We will define the integral for ‘integrable‘ functions. We don’t know what this means yet but thisContinue reading “Almost everywhere – the new everywhere”

## Fatou’s lemma and more properties of lebesgue’s integral

Now that we are equipped with the monotone convergence theorem we can derive some interesting results from it. The first I want to present is Fatou’s lemma – a weaker version of the theorem that deals with non-monotonic sequences! What is Fatou’s lemma? So we’ve seen that the monotone convergence theorem only works with… monotonicContinue reading “Fatou’s lemma and more properties of lebesgue’s integral”

## Lebesgue’s Monotone convergence theorem

After all the praises Lebesgue’s Monotone convergence theorem got in the last post, it’s now finally time to present it, prove it, and reap it’s benefits! Let’s do it! The Theorem Suppose that for every the function is defined and measurable. Moreover, suppose that for every : Then the function: Is also defined and measurable.Continue reading “Lebesgue’s Monotone convergence theorem”

## Lebesgue integral of non-negative functions

In the last post we’ve defined the lebesgue integral for indicators – functions of the form: for some measurable set . We’ve defined the integral to be the measure of the set . After doing so, we’ve defined the integral for a bit more comlicated functions, simple functions. However, simple functions are just a linearContinue reading “Lebesgue integral of non-negative functions”

## Indicators & simple functions – Lebesgue integral

After we’ve seen what measurable sets, measurable functions and measurable spaces are We are finally ready to define the integral. Yes, the integral I’ve been talking about since the first post! But, this is not going to be so easy. The construction of the integral is going to be done in 4 parts. The firstContinue reading “Indicators & simple functions – Lebesgue integral”

## Measurable functions & Measure Spaces

So far, all of what I’ve been talking about was concerning the real line. The measurable sets that I’ve defined were subsets of , there is a good for reason for that though: If you remember, our main goal since the first post was to give a better definition to an Integral. However, in mathContinue reading “Measurable functions & Measure Spaces”

## Measurable sets – Part 2

I am going to start now exactly where I stopped in the last post. I will prove that the set of measurable sets is indeed a -algebra. To do so, I only need to prove that if is a collection of measurable sets, then is also a measurable set. First I want to prove aContinue reading “Measurable sets – Part 2”

## Measurable sets

In the last post, we were seeking for an ‘ideal measure function, however, we found out that such a function does not exits. There were four properties we wanted from the function to fulfill: For every , is defined and satisfies For every interval , . If we ‘shift’ / ‘slide’ a set, we don’tContinue reading “Measurable sets”

## Pursuing the ideal measure

In the last post, I have listed 4 properties I want an ideal measure on the real line to fulfill. The properties were: For every , is defined and satisfies For every interval , . If we ‘shift’ / ‘slide’ a set, we don’t want the output of the measure to change. If we considerContinue reading “Pursuing the ideal measure”