For $A\subset X$, prove that the characteristic function $\chi_A$ (which maps from $X\to\{0,1\}$) is $M$-measurable $\iff$ $A\in M$.

For $A\subset X$, prove that the characteristic function $\chi_A$ (which
maps from $X\to\{0,1\}$) is $M$-measurable $\iff$ $A\in M$.

For $A\subset X$, prove that the characteristic function $\chi_A$ (which
maps from $X\to\{0,1\}$) is $M$-measurable $\iff$ $A\in M$.
I got the forward direction (i.e, proving $A\subset M$). For the other
direction, if $A\subset M$, then $\forall t\ge1$, $f^{-1}([-\infty,t])=X$.
But why is it true that $X\in M$ (this must be satisfied for the function
to be $M$-measurable)? Thank you.