Integration of Non-Negative Measurable Functions: Fatou's Lemma, the Monotone Convergence Theorem, and Beppo Levi's Theorem

Let $\left\{ u_{n}: X \rightarrow [0, \infty ]\right\}$ be a sequence of measurable functions. For $x \in X$, define $f(x) = \sum_{n = 1}^{\infty } u_{n}(x)$. Show that

Show that if $f: X \rightarrow [0, \infty ]$ is measurable, then

Let $g: X \rightarrow [0, \infty ]$ be integrable. Define

Let $f: X \rightarrow [0, \infty ]$ and $g: X \rightarrow [0, \infty ]$ be measurable functions for which $f^{2}$ and $g^{2}$ are integrable. Show that $f \cdot g$ also is integrable.

Let $X$ be the union of a countable, ascending sequence of measurable sets $\left\{ X_{n}\right\}$ and $f: X \rightarrow [0, \infty ]$ be measurable. Show that $f$ is integrable over $X$ if and only if there is an $M \geq 0$ for which $\int_{X_{n}} f d \mu \leq M$ for all $n$.

Let $\eta$ be the counting measure on $2^{\mathrm{N}}$. Characterize the non-negative, real-valued functions (that is, sequences) that are integrable over $\mathbb {N}$ with respect to $\eta$ and the value of $\int_{\mathbb {N}} f d \eta$.

Let $x_{0} \in X$ and $\delta_{x_{0}}$ be the Dirac measure concentrated at $x_{0}$. Characterize the non-negative, real-valued functions on $X$ that are integrable over $X$ with respect to $\delta_{x_{0}}$ and the value of $\int_{X} f d \delta_{x_{0}}$.