Problem 2
Prove that if each of the measures \(\mu_{1}\) and \(\mu_{2}\) has a countable base, then so does their direct product \(\mu=\mu_{1} \times \mu_{2}\). Comment. In particular, Lebesgue measure in the plane (or more generally in n-space) has a countable base.
Problem 4
Prove that \(L_{2}(X, \mu)\) is separable, i.e., has a countable everywhere dense subset, if \(\mu\) has a countable base. Comment. Thus \(L_{2}(X, \mu)\) is a Hilbert space if \(\mu\) has a countable base (we disregard the case where \(L_{2}(X, \mu)\) is finite-dimensional). It follows from Theorem 11 , p. 155 that all such spaces are isomorphic, in particular, that \(L_{2}(X, \mu)\) is isomorphic to the space \(l_{2}\) of all sequences \(\left(x_{n}, x_{n}, \ldots, x_{\prime}, \ldots\right)\) such that $$ \sum_{n=1}^{\infty} x_{n}^{2}<\infty $$ (in fact, \(l_{2}\) corresponds to the case where the measure \(\mu\) is concentrated on a countable set of points).
Problem 9
Give an example of a sequence of functions \(\left\\{f_{n}\right\\}\) which converges everywhere on \([0,1]\), but does not converge in the mean. Hint. Let $$ f_{n}(x)= \begin{cases}n & \text { if } \quad x \in(0,1 / n) \\ 0 & \text { otherwise }\end{cases} $$
Problem 10
Give an example of a sequence of functions (f) which converges uniformly, but does not converge in the mean or in the mean square. Hint. According to Problem \(7 \mathrm{a}\), we must have \(\mu(X)=\infty .\) Let $$ f_{n}(x)= \begin{cases}\frac{1}{\sqrt{n}} & \text { if } \quad|x| \leq n \\ 0 & \text { if } \quad|x|>n\end{cases} $$
Problem 12
Let \(L_{p}(X, \mu)\) be the set of all classes of equivalent (real or complex) functions \(\mathbf{f}\) such that $$ \int|f|^{n} d \mu<\infty \quad(1
