1
$\begingroup$

Let $X_n(w) = nI_{(n-1,n)}(w)$ be a function on the real numbers with the Borel sigma algebra. Consider Lebesgue measure, denoted by $\mu$. Can this be a random variable? Does this converge to 0 almost surely?This is equivalent to showing $f_n(x) = nI_{(n-1,n)}(x) \rightarrow 0 \ a.e.[\mu]$. What about pointwise? How do you prove this?

$\endgroup$
0
$\begingroup$

The expression "random variable" usually mean a function defined on a probability space, which is not the case of the real line endowed with Lebesgue measure because it has an infinite measure.

We can view the sequence $(X_n)$ as a sequence of measurable functions. Since the intervals $(n-1,n)$ are pairwise disjoint, we have for a fixed $\omega$ that $X_n(\omega)=1$ for at most one $n$. Hence the sequence converges pointwise to $0$. In particular it converges almost everywhere for any measure (probability measure or not).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.