Measurability of a set in the definition of almost sure convergence

$$A=\{\omega\in\Omega\mid\text{ for all }\epsilon>0\text{ there is } N\text{ such that } |X_n(\omega)-X(\omega)|<\epsilon\text{ for all }n\geq N\}$$ $$=\{\omega\in\Omega\mid\text{ for all }m\text{ there is } N\text{ such that } |X_n(\omega)-X(\omega)|<1/m\text{ for all }n\geq N\}$$ $$=\bigcap_{m=1}^\infty\bigcup_{N=1}^\infty\bigcap_{n=N}^\infty\{\omega\in\Omega\mid|X_n(\omega)-X(\omega)|<1/m\}.$$

Assuming $X$ is real-valued, $$\{ \omega \in \Omega: \lim_{n \rightarrow \infty} X_n(\omega) = X(\omega)\} = ((X-\limsup_n X_n)=0)\cap ((X-\liminf_n X_n)=0) $$

Since $\limsup_n X_n$, $\liminf_n X_n$ and $X$ are measurable, $((X-\limsup_n X_n)=0)\cap ((X-\liminf_n X_n)=0)$ is measurable.