Let $(\Omega, \mathcal{A}, \mu)$ be a measure space and $f: \Omega \rightarrow \mathbb{R}$ a measurable function.

(a) Explain what is meant by saying that $f$ is integrable, and how the integral $\int_{\Omega} f d \mu$ is defined, starting with integrals of $\mathcal{A}$-simple functions.

[Your answer should consist of clear definitions, including the ones for $\mathcal{A}$-simple functions and their integrals.]

(b) For $f: \Omega \rightarrow[0, \infty)$ give a specific sequence $\left(g_{n}\right)_{n \in \mathbb{N}}$ of $\mathcal{A}$-simple functions such that $0 \leqslant g_{n} \leqslant f$ and $g_{n}(x) \rightarrow f(x)$ for all $x \in \Omega$. Justify your answer.

(c) Suppose that that $\mu(\Omega)<\infty$ and let $f_{1}, f_{2}, \ldots: \Omega \rightarrow \mathbb{R}$ be measurable functions such that $f_{n}(x) \rightarrow 0$ for all $x \in \Omega$. Prove that, if

$$\lim _{c \rightarrow \infty} \sup _{n \in \mathbb{N}} \int_{\left|f_{n}\right|>c}\left|f_{n}\right| d \mu=0$$

then $\int_{\Omega} f_{n} d \mu \rightarrow 0$.

Give an example with $\mu(\Omega)<\infty$ such that $f_{n}(x) \rightarrow 0$ for all $x \in \Omega$, but $\int_{\Omega} f_{n} d \mu \nrightarrow 0$, and justify your answer.

(d) State and prove Fatou's Lemma for a sequence of non-negative measurable functions.

[Standard results on measurability and integration may be used without proof.]