91Ó°ÊÓ

Show that \(\xi=\xi(\omega)\) is an extended random variable if and only if \(\\{\omega: \xi(\omega) \in \bar{B}\\} \in F\) for all \(\boldsymbol{B} \in \mathscr{S}(\boldsymbol{R})\).

Show that a distribution function \(F=F(x)\) on \(R\) has at most a countable set of points of discontinuity. Does a corresponding result hold for distribution functions on \(R^{n}\) ?

Using Fatou's lemma, show that $$ \mathrm{P}\left(\underline{\lim } A_{n}\right) \leq \varliminf \mathrm{P}\left(A_{n}\right), \quad \mathrm{P}\left(\overline{\lim } A_{n}\right) \geq \varlimsup \mathrm{lim} \mathrm{P}\left(A_{n}\right) $$

Let \((\Omega, F, P)\) be a probability space and \(\mathcal{A}\) an algebra of subsets of \(\Omega\) such that \(o(\mathscr{A})=\mathscr{F}\). Using the principle of appropriate sets, prove that for every \(\varepsilon>0\) and \(B \in F\) there is a set \(A \in \mathscr{A}\) such that $$ P(A \triangle B) \leq \varepsilon. $$

Dirichlet's function $$ d(x)= \begin{cases}1, & x \text { irrational, } \\ 0, & x \text { rational, }\end{cases} $$ is defined on \([0,1]\), Lebesgue integrable, but not Riemann integrable. Why?