# Is the point-wise limit of simple functions a measurable function?

Let $$X$$ and $$Y$$ be topological spaces. By a simple function $$\phi: X\to Y$$ we mean a finite range Borel measurable function.

Q. Is the point-wise limit of a sequence of simple functions a Borel measurable function?

• This question is better suited for MSE, in fact it´s answered here Jan 17, 2022 at 14:24
• @Saúl Rodríguez Martín, It is concerned with set-valued functions not complex valued! I have no clue to find a proof
– ABB
Jan 17, 2022 at 14:38
• Your notation is unknown to me. What is the definition of a finite range Borel measurable function? Jan 17, 2022 at 15:15
• In fact the standard definition is actually: simple function=finite range, Borel measurable function=linear combination of characteristic functions of Borel subsets. But where are the set-valued functions? Jan 17, 2022 at 21:07
• @PietroMajer I was confused at first too, but I think he means that the function takes values in any set $Y$, not just in $\mathbb{C}$ (although that´s not what people usually mean when they say set-valued). Jan 18, 2022 at 22:16

The answer to your question is Yes provided the topology of $$Y$$ is such that for each non-empty open set $$O\subset Y$$ there is a strictly increasing sequence $$(O_k)$$ of open sets: $$\overline O_k\subset O_{k+1}\subset O,\quad k=1,2,\ldots,$$ with $$\bigcup_kO_k=O$$. For suppose $$(f_n)$$ is a sequence of Borel functions from $$X$$ to $$Y$$ with pointwise limit $$f$$. With $$O$$ and the $$O_k$$ as above, $$f^{-1}(O)=\bigcup_{k=1}^\infty\bigcup_{n=1}^\infty\bigcap_{m=n}^\infty f_m^{-1}(O_k).$$ This shows that $$f^{-1}(O)$$ is a measurable subset of $$X$$. Because the Borel $$\sigma$$-field on $$Y$$ is generated by the open sets, it follows that $$f$$ is Borel measurable.
In particular, the condition above is true if $$Y$$ is metrizable.
• I do not think that you need a strictly increasing sequence with strict inclusions. And in fact that "strict" need not hold in metrizable spaces (e.g. if $Y$ is finite). Jan 18, 2022 at 17:37
• I tried to indicate what I meant by strictly in the display just below that word. In particular, my use of $A\subset B$ follows the convention allowing for $A$ and $B$ to coincide. Jan 18, 2022 at 18:36