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Let $f_n:X\rightarrow\mathbb{R}$, $n\in\mathbb{N}$ be a sequence of measurable functions. Show that the set of points, $x\in X$, where the sequence $\{f_n(x)\}$ converges is measurable. Then deduce that the set of points, $\omega\in I$, for which the "randomized harmonic series" $\sum\limits_{n=1}^\infty\frac{R_n(\omega)}{n}$ converges is a measurable subset of $I$.

So I know the definition of a measurable function is: for all $a\in\mathbb{R}$, the set $\{x\in X; f(x)>a\}$ is an element of the field. Also, a measurable set corresponds to a Borel set. I just don't know how to get from one to the other.

I've seen the first part of this question posted (Set of points where sequence converges is measurable), but it wasn't very helpful for me.

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The limit of a monotone sequence of measurable functions is still measurable. Now $\{x\in X:\lim_{n}f_{n}(x)~\text{exists}\}=\{x\in X: \limsup_{n}f_{n}(x)=\liminf_{n}f_{n}(x)\}$. Note that $\limsup_{n}f_{n}=\lim_{n}\sup_{k\geq n}f_{k}$ and $\liminf_{n}f_{n}=\lim_{n}\inf_{k\geq n}f_{k}$.

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