Let $(E, \mathcal{E}, \mu)$ be a measure space. Explain what is meant by a simple function on $(E, \mathcal{E}, \mu)$ and state the definition of the integral of a simple function with respect to $\mu$.

Explain what is meant by an integrable function on $(E, \mathcal{E}, \mu)$ and explain how the integral of such a function is defined.

State the monotone convergence theorem.

Show that the following map is linear

$$f \mapsto \mu(f): L^{1}(E, \mathcal{E}, \mu) \rightarrow \mathbb{R}$$

where $\mu(f)$ denotes the integral of $f$ with respect to $\mu$.

[You may assume without proof any fact concerning simple functions and their integrals. You are not expected to prove the monotone convergence theorem.]