Explain what is meant by a simple function on a measurable space $(S, \mathcal{S})$.

Let $(S, \mathcal{S}, \mu)$ be a finite measure space and let $f: S \rightarrow \mathbb{R}$ be a non-negative Borel measurable function. State the definition of the integral of $f$ with respect to $\mu$.

Prove that, for any sequence of simple functions $\left(g_{n}\right)$ such that $0 \leqslant g_{n}(x) \uparrow f(x)$ for all $x \in S$, we have

$$\int g_{n} d \mu \uparrow \int f d \mu .$$

State and prove the Monotone Convergence Theorem for finite measure spaces.