Show a step function is measurable

Question

Suppose $X_n$ is a step function that converges pointwise to $X$, where $X: \Omega \rightarrow \mathbb{R} $ is a measurable function. How would I show that that $X_n$ is measurable with respect to $\sigma (X)$?

Answer 1

It is not true in general.

Let $X$ be constant so that $\sigma(X)=\{\varnothing,\Omega\}$.

Then it is not difficult to find a sequence of step functions converging to $X$, but these functions are not necessarily constant.

However every function that is measurable wrt $\{\varnothing,\Omega\}$ is constant.