What does it mean to be a measurable function from a measurable set?

Question

let $(X,\Sigma)$ be a measure space, and let $A\in \Sigma$. What does it mean to say that $f:A\to\mathbb{R}$ is measurable?

what is the $\sigma$-algebra on A?

Answer 1

Since $A\in \Sigma$, the $ \sigma - $ algebra on $A$ is given by

$$\{B \in \Sigma: B \subseteq A\}.$$