# Measurability/Lebesgue Integration

• Ted123
In summary, the student is asking for help with part (c) of a question involving integration. They have already solved part (a) using Wolfram Alpha and are looking for a way to integrate the equation by hand. They are also unsure about their approach in part (c) and are seeking clarification. The expert suggests a method for integrating the equation by hand and confirms that the student's approach in part (c) is correct.
Ted123

For part (a):

## The Attempt at a Solution

I think I've done all of this question except for the very last part of (c).

For (a) I've found the integral $$\int \frac{x^2-y^2}{(x^2+y^2)^2}\;dx$$ using Wolfram Alpha but how would I integrate it bare handedly?

Since the double integral wrt x then y does not equal the double integral wrt y then x, f is not Lebesgue integrable over [0,1]x[0,1]. However if 0<a<b then f is Lebesgue integrable over [a,b]x[a,b] and the integral is 0 by Fubini.

The last part of (c) is what I'm not sure about: deducing that $$\int f = |S^+| - |S^-|.$$ Am I going about it the right way? We can write $f=f^+ - f^-$ where $f^+ = \max (f,0)$ and $f^- = -\min (f,0)$ (i.e. the +ve and -ve parts of f respectively).

Then we know if $f\in L^1 (\mathbb{R})$ then $f^+ , f^- \in L^1 (\mathbb{R})$ and $f^+ , f^- \geq 0$.

Now is the following correct? $$\int f = \int (f^+ - f^-) = \int f^+ - \int f^- = |S^+| - |S^-|$$

Ted123 said:
For (a) I've found the integral $$\int \frac{x^2-y^2}{(x^2+y^2)^2}\;dx$$ using Wolfram Alpha but how would I integrate it bare handedly?
If guess you could notice that it might be d/dx of something over (x^2+y^2) (because of the quotient rule), and then guess what the numerator should be.

Or, you could proceed as follows: \begin{align*} \int \frac{x^2-y^2}{(x^2+y^2)^2}\;dx &= \int \frac{x^2}{(x^2+y^2)^2}\;dx - y^2 \int \frac{1}{(x^2+y^2)^2}\;dx \\ &= \int \frac{1}{x^2+y^2} - \frac{y^2}{(x^2+y^2)^2}\;dx - y^2 \int \frac{1}{(x^2+y^2)^2}\;dx \\ &= \int \frac{dx}{x^2+y^2} - 2y^2 \int \frac{1}{(x^2+y^2)^2}\;dx. \end{align*}

To deal with the second integral on the RHS, all you need to know is how to integrate
$$\int \frac{dv}{(v^2+1)^2}.$$ But this is easy (though slightly tedious). E.g. start off with the sub v=tan(t).

The last part of (c) is what I'm not sure about: deducing that $$\int f = |S^+| - |S^-|.$$ Am I going about it the right way? We can write $f=f^+ - f^-$ where $f^+ = \max (f,0)$ and $f^- = -\min (f,0)$ (i.e. the +ve and -ve parts of f respectively).

Then we know if $f\in L^1 (\mathbb{R})$ then $f^+ , f^- \in L^1 (\mathbb{R})$ and $f^+ , f^- \geq 0$.

Now is the following correct? $$\int f = \int (f^+ - f^-) = \int f^+ - \int f^- = |S^+| - |S^-|$$
Yes, of course. This follows from the previous part applied to ##f^+## and ##f^-##.

## 1. What is measurability in Lebesgue integration?

Measurability in Lebesgue integration refers to the property of a set or a function to be measurable with respect to a given measure. In other words, it is the ability to assign a number to a set or function that represents its size or extent.

## 2. What is the difference between Riemann and Lebesgue integration?

The main difference between Riemann and Lebesgue integration is in the way they define and evaluate the integral of a function. Riemann integration uses the concept of partitions and limits of sums, while Lebesgue integration uses the concept of measure and sets of measure zero.

## 3. Why is Lebesgue integration important in mathematics?

Lebesgue integration is important in mathematics because it provides a more general and powerful framework for integration compared to Riemann integration. It allows for the integration of a wider class of functions and has better convergence properties, making it useful in various areas of mathematics and physics.

## 4. What is the Lebesgue measure?

The Lebesgue measure is a way of assigning a number to a set in order to represent its size or extent. It is defined as the infimum of the sum of the volumes of a countable collection of open intervals that cover the set. It is used in Lebesgue integration to determine the size of sets and to define integrals of functions.

## 5. How is Lebesgue integration used in probability theory?

In probability theory, Lebesgue integration is used to define the concept of probability measures, which are used to assign probabilities to events. It also allows for the integration of random variables, which are used to model probabilistic phenomena. Lebesgue integration is an essential tool in the study of probability and statistics.

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