TITLE: measurability of the limit of functions parameterized by real numbers
QUESTION [1 upvotes]: Let $f_s: \mathbb R \to \mathbb R$ a be family of Borel measurable functions  parameterized by $s\in \mathbb R$. Consider the limit function 
$$ F(t)=\limsup_{s\to 0} f_s(t).  $$ Is the function $F$ Borel measurable. 
This seems to be not true in general. 
Consider a locally finite Borel measure $\mu$ on $\mathbb R$. Is the function 
$$F(x)=\liminf _{r\to 0+} \frac{log (\mu([x-r, x+r]))}{\log r}  $$
Borel measurable?

REPLY [1 votes]: The function $F$ need not be Borel, even if $(s,t) \mapsto f_s(t)$ is a Borel function on $\mathbb{R}^2$.  I wrote down a counterexample on Math.SE.
{"set_name": "stack_exchange", "score": 1, "question_id": 282968}