Convergence theorems for measurable functions

The swapping of integration and taking limit of the integrand, like \(\int \lim_{n \rightarrow \infty} f_n = \lim_{n \rightarrow \infty} \int f_n \), is usually taken for granted as a valid operation in engineering courses. However, if we require mathematical rigorousness, such manipulation relies on additional constraints in order to be feasible. This is governed by a set of convergence theorems about measurable functions. By referring to Halsey Royden's "Real Analysis (3th ed., 2004)", this article compiles these theorems to present an overview of their similarities and differences.

Let \(\{f_n\}_{n \geq 1}\) be a sequence of measurable functions defined on a measurable set \(E\). The measure of \(E\) is \({\rm m}(E)\). Let \(g\) be integrable on \(E\) and \(\{g_n\}_{n \geq 1}\) be a sequence of integrable functions which converges a.e. to \(g\). The integrals in the following are in the sense of Lebesgue integral.

Theorem	Requirements on $$\{f_n\}_{n \geq 1}$$	$${\rm m}(E)$$	Convergence of $$\{f_n\}_{n \geq 1}$$	Boundedness of $$\{f_n\}_{n \geq 1}$$	Swapping of integration and taking limit
Bounded convergence theorem	Measurable	$${\rm m}(E)\in [0, \infty)$$	$f_n \rightarrow f$ on $E$	$$\abs{f_n(x)} \leq M$$	\[\displaystyle{\int_E f = \lim_{n \rightarrow \infty} \int_E f_n}\]
Fatou's Lemma	1. Measurable 2. Nonnegative	$${\rm m}(E) \in [0,\infty]$$	$f_n \rightarrow f$ a.e. on $E$	None	\[\displaystyle{\int_E f \leq \underline \lim\int_E f_n}\]
Monotone convergence theorem	1. Measurable 2. Nonnegative 3. Increasing	$${\rm m}(E) \in [0,\infty]$$	$f_n \rightarrow f$ a.e. on $E$	None	\[\displaystyle{\int_E f = \lim_{n \rightarrow \infty} \int_E f_n}\]
Lebesgue convergence theorem	Measurable	$${\rm m}(E) \in [0,\infty]$$	$f_n \rightarrow f$ a.e. on $E$	1. $\abs{f_n} \leq g$ 2. $\int_E g < \infty$	\[\displaystyle{\int_E f = \lim_{n \rightarrow \infty} \int_E f_n}\]
Extended Lebesgue convergence theorem	Measurable	$${\rm m}(E) \in [0,\infty]$$	$f_n \rightarrow f$ a.e. on $E$	1. $\abs{f_n} \leq g_n$ 2. $\int_E g_n < \infty$ 3. $g_n \rightarrow g$ a.e. on $E$ 4. $\int_E g < \infty$ 5. $\int_E g = \lim_{n \rightarrow \infty} \int_E g_n$	\[\displaystyle{\int_E f = \lim_{n \rightarrow \infty} \int_E f_n}\]