# Taking away the “almost sure” [closed]

Given an arbitrary sequence of random variables (or say measurable functions on a finite-measure space) $\xi_n$, one can show by a truncation and Borel-Cantelli argument that there always exists a sequence $c_n>0$ such that $$\sum_{n=1}^\infty c_n \xi_n \quad \text{converges almost surely.}$$

Can one give an example to show that the "almost surely" in the statement CANNOT be strengthened to "pointwise"?

• RVs in general are not defined pointwise EVERYWHERE, so this question does not make a lot of sense. – Asaf Oct 5 '16 at 20:21
• To make sense of the question, you might say: find a sequence of real-valued Borel functions $\xi_n$ on, say, $[0,1]$, such that there is no sequence $c_n > 0$ for which $\sum_{n=1}^\infty c_n \xi_n$ converges pointwise. – Robert Israel Oct 5 '16 at 20:24

## 1 Answer

Let $\Omega=(0,1)$ and $\xi_n:\omega\in\Omega\mapsto$ the $n$-th term of the continued fraction expansion of $\omega$. Given a sequence $c_n$, there is another sequence $m_n\in\mathbb{N}$ such that $\sum_{n=1}^\infty c_nm_n$ diverges. Let $x=[m_1,m_2,\dots,m_n,\dots]$. Then $\sum_{n=1}^\infty c_n\xi_n(x)$ diverges.