Prove that a measurable function is monotone.

Question

I don't really know how to approach tis question. I have done the first part, but I am struggling with part b. The question goes like this...

Let $(X, \Sigma, \mu)$ be a measure space.

a) Assume the set $L_c=\{x\in X:f(x) < c\}$ belongs to $\Sigma$ for every $c \in \mathbb{R}$. Prove that $f$ is a measurable function.

b) Denote now $g(t)=\mu(L_t)$, where $L_t \in \Sigma$ is defined in (a). Prove that $g:\mathbb{R}\rightarrow [0,\infty) $ is a monotone function. Using this or otherwise show that $\{t:\mu(L_t) < 10\}$ is a Borel subset of $\mathbb{R}$.

Answer 1

For 1), show that the collection of sets $A \subset \mathbb{R}$ such that $f^{-1}(A)$ is measurable contains all of the open sets and is a $\sigma$-algebra. This shows that it contains the Borel sets and hence $f$ is measurable.

For 2) use the fact that $\mu(A) \le \mu(B)$ whenever $A \subset B$ to conclude that $g$ is monotone.