91Ó°ÊÓ

Define \(\mu^{*}(E)\) as the number of points in \(E\) if \(E\) is finite and \(\mu^{*}(E)=\infty\) if \(E\) is infinite. Show that \(\mu^{*}\) is an outer measure. Determine the measurable sets.

A sequence \(\left\\{x_{x}\right\\}\) is convergent to \(y\) if and only if every subsequence \(\left\\{x_{n_{*}}\right\\}\) is convergent to \(y\).

If \(X=\bigcup_{n=1}^{\infty} A_{n}\) and \(\mu\) is a signed measure with \(\left|\mu\left(A_{n}\right)\right|<\infty\) for all \(n\), then \(\mu^{+}\)and \(\mu^{-}\)are \(\sigma\)-finite. (Hence, by definition, \(\mu\) is \(\sigma\)-finite.)

Let \(a_{i}

The Lebesgue measure of a point is zero.

Let \(\mu\) be a complete measure. A set for which \(\mu(N)=0\) is called a null set. Show that the class of nuli scts is a \(\sigma\)-ring. Is it also a \(\sigma\)-algebra?

The Lebesgue measure of a countable set of points is zero.

Let \(\mu\) be a measure with domain \(Q\). For any two sets \(E\) and \(F\) in \((Q\), let \(\rho(E, F)=\mu[(E-F) \cup(F-E)]\). Prove that \(\rho(E, F)=\rho(F, E)\) and \(p(E, G) \leq \rho(E, F)+\rho(F, G)\).

Define \(\mu^{*}(\varnothing)=0, \mu^{*}(E)=1\) if \(E \neq \varnothing\). Show that \(\mu^{*}\) is an outer measure, and determine the measurable sets.

Let \(X\) have a noncountable number of points. Set \(\mu^{*}(E)=0\) if \(E\) is countable, \(\mu^{*}(E)=1\) if \(E\) is noncountable. Show that \(\mu^{*}\) is an outer measure, and determine the measurable sets.