Problem 1
Prove that the Dirichlet function $$ f(x)= \begin{cases}1 & \text { if } \mathrm{x} \text { is rational } \\ 0 & \text { if } x \text { is irrational }\end{cases} $$ is measurable on every interval \([a, b]\).
Problem 4
Prove that iff is measurable, then so is \(|f|\).
Problem 5
Let \(\left\\{f_{n}\right\\}\) be a sequence of measurable functions converging almost everywhere to a functionf. Prove that \(\left\\{f_{n}\right\\}\) converges almost everywhere to a function \(g\) if and only if \(f\) and \(g\) are equivalent.
Problem 9
Prove that if a sequence \(\left\\{f_{n}\right\\}\) of functions converges to \(\mathbf{f}\) in measure, then it contains a subsequence \(\left\\{f_{n_{k}}\right\\}\) converging to \(\mathbf{f}\) almost everywhere. Hint. Let \(\left\\{\delta_{n}\right\\}\) be a sequence of positive numbers such that $$ \lim _{n \rightarrow \infty} \delta_{n}=0 $$ and let \(\left\\{\varepsilon_{n}\right\\}\) be a sequence of positive numbers such that $$ \sum_{n=1}^{\infty} \varepsilon_{n}<\infty $$ Let \\{n) be a sequence of positive integers such that \(\mathrm{n}\), \(>n_{k-1}\) and $$ \mu\left\\{x:\left|f_{n_{k}}(x)-f(x)\right|>\delta_{k}\right\\}<\varepsilon_{k} \quad(k=1,2, \ldots) $$ Moreover, let $$ R_{i}=\bigcup_{k=f}^{\infty}\left\\{x:\left|f_{n_{k}}(x)-f(x)\right|>\delta_{k}\right\\}, \quad Q=\bigcap_{i=1}^{\infty} R_{i} $$ Then \(\mu\left(R_{i}\right) \rightarrow \mu(Q)\) as \(\mathbf{i} \rightarrow \infty\), since \(R_{1} \supset R_{2} \supset \cdot \ldots\) On the other hand, $$ \mu\left(R_{i}\right)<\sum_{k=1}^{m} \varepsilon_{k} $$ and hence \(\mu\left(R_{i}\right) \rightarrow 0\), so that \(\mu(Q)=0\). Now show that \(\left\\{f_{n_{n}}\right\\}\) converges to \(f\) on \(E-Q .\)
Problem 10
Prove that a function \(f\) defined on a closed interval [a, \(b]\) is p-measurable if and only if, given any \(\varepsilon>0\), there is a continuous function \(\varphi\) on \([a, b]\) such that \(\mu\\{x: f(x) \neq \varphi(x)\\}<\mathrm{E}\) Hint. Use Egorov's theorem. Comment. This result, known as Luzin's theorem, shows that a measurable function "can be made continuous by altering it on a set of arbitrarily small measure."
