Example of limit of a net of Borel functions not Borel

Question

I am stuck in the following problem on Reed and Simon functional analysis.

Give an example to show that a pointwise limit of a net of Borel functions on $\mathbb{R}$ may not be Borel.

First of all, that net cannot be a sequence in the usual sense, because from real analysis we have pointwise sequence limit of Borel functions is Borel.

I try to find some non-Borel measurable function, like the characteristic function of this. Then I try to imagine if it could be written as limit of Borel function in some sense. However I'm not sure what type of net should I look for, and how to characterize the convergence of such a net.

Answer 1

Let $A$ be a subset of $\mathbb R$ that is not Borel. I'll produce a net of Borel functions on $\mathbb R$ whose limit is the characteristic function of $A$. The index set $I$ for my net is the set of finite subsets of $A$, and the partial ordering of $I$ is the inclusion relation $\subseteq$. This partial order is directed because the union of any two finite subsets of $A$ is a finite subset of $A$. The net $(f_i)_{i\in I}$ is defined by letting $f_i$ be the characteristic function of the finite set $i$. Each $f_i$, being the characteristic function of a Borel set (in fact, of a finite set), is a Borel function. I claim that the pointwise limit of $(f_i)_{i\in I}$ is the characteristic function of $A$. Indeed, if $x\in A$ then $f_i(x)=1$ for all $i\geq\{x\}$ in $I$, and, on the other hand, if $x\notin A$ then $f_i(x)=0$ for all $i\in A$.