The Integral of a Non-Negative Measurable Function

Let $\left\{ E_{n}\right\}_{n = 1}^{\infty }$ be a measurable partition of $E$. Just using the Monotone Convergence Theorem and the linearity of integration, show that $m(E) = \sum_{n = 1}^{\infty } m\left(E_{n}\right)$.

If $f:[a, b] \rightarrow [0, \infty ]$ is measurable, show that $\int_{[a, b]} f d m = \int_{(a, b)} f d m$.

Let $\left\{ u_{n}: E \rightarrow [0, \infty ]\right\}$ be a sequence of measurable functions. Define $s = \sum_{n = 1}^{\infty } u_{n}$. Show that $\int_{E} s d m = \sum_{n = 1}^{\infty } \int_{E} u_{n} d m$.

If $f:[a, b] \rightarrow [0, \infty ]$ is measurable, show that $\int_{[a, b]} f d m = 0$ if and only if $f = 0$ almost everywhere

Prove the linearity of integration for general coefficients $\alpha$ and $\beta$.

Let $\left\{ a_{n}\right\}$ be a sequence of non-negative real numbers. Define the function $f$ on $E = [1, \infty )$ by setting $f(x) = a_{n}$ if $n \leq x < n+1$. Show that $\int_{E} f d m = \sum_{n = 1}^{\infty } a_{n}$.

Let $f: E \rightarrow [0, \infty ]$ be measurable. Show that

where the supremum is taken over finitely supported, simple functions.

Let $\left\{ f_{n}: E \rightarrow [0, \infty ]\right\}$ be a sequence of measurable functions that converges pointwise on $E$ to $f$. Suppose $f_{n} \leq f$ on $E$ for each $n$. Show that $\lim_{n \rightarrow \infty } \int_{E} f_{n} d m = \int_{E} f d m$.

Show that the Monotone Convergence Theorem does not hold for decreasing sequences of functions.

Prove the following generalization of Fatou's Lemma: If $\left\{ f_{n}: E \rightarrow [0, \infty ]\right\}$ is a sequence of measurable functions, then $\int_{E}\left[\liminf f_{n}\right] d m \leq \liminf \int_{E} f_{n} d m$.