Firstly,
Definition 1: function f is measurable if we have a sequence of simple function $s_n$ such that $s_n \to f$.
Definition 2: a set $A$ is measurable if characteristic function $\chi_A$ is measurable.
but by these definition, I very confuse when I prove that continuous function then measurable. Can anyone give me a proof?.
PS: only use these definition.
Secondly, I have Another question: show that if $A_n$ are measurable sets then $\bigcap_{n=1}^{\infty}A_n$ also measurable set.
My solution: Beacause $A_n$ are measurable sets, characteristic functions $\chi_{A_n}$ measurable, or exist sequence of simple finction $s_n$ s.t $s_n \to \chi_{A_n}$. So $s=\prod_{n=1}^{\infty}s_n \to \prod_{n=1}^{\infty}\chi_{A_n}=\chi_{\bigcap_{n=1}^{\infty}A_n}$ and since $s$ also simple function, we have $\chi_{\bigcap_{n=1}^{\infty}A_n}$ measurable. Therefore $\bigcap_{n=1}^{\infty}A_n$ measurable. QED
I still feel something is wrong. Can anyone help me check my solution?. Thankful