Measurability of a map constructed by a family of measurable function

Question

Assume that $(X, \mathcal{B}_X,\mu)$ be a Borel measurable probability space where $X$ is complete separable metric space.

Let $\{T_x\}_{x\in X}$ be a family of measurable maps $T_x:X\to [0,1]$. Note that this family is indexed by the space $X$. Define $$ \mathcal{T}:X\to[0,1],\quad \mathcal{T}(x)=T_x(x).$$

Question: Is the map $\mathcal{T}$ measurable?

I do not how to approach this problem and if being complete separable metric space is useful or not.

Thank you in advance.

Answer 1

Let $X$ be the real line, $E$ a subset of $\Delta=\{(x,y): x=y\}$ which is not a Borel set . Let $T_x(y)=I_E(x,y)$. Then $T_x$ is a constant for each $x$, hence measurable, but $x \mapsto T_x(x)$ is not measurable.