Convergence Theorems for Integrals Review

# Convergence Theorems for Integrals Review

We will now review all of the convergence theorems for integrals that we have looked at thus far.

Theorem Hypotheses Conclusion
Lebesgue Integrability of Functions that are 0 Almost Everywhere 1) $f(x) = 0$ almost everywhere on $I$. a) $f \in L(I)$.
b) $\displaystyle{\int_I f(x) \: dx = 0}$.
Lebesgue Integrability of Functions Equalling Lebesgue Integrable Functions Almost Everywhere 1) $f(x) = g(x)$ almost everywhere on $I$.
2) $f \in L(I)$
a) $g \in L(I)$.
b) $\displaystyle{\int_I f(x) \: dx = \int_I g(x) \: dx}$.
Levi's Monotone Convergence Theorem for Step Functions 1) $(f_n(x))_{n=1}^{\infty}$ is a sequence of step functions that is increasing almost everywhere on $I$.
2) $\displaystyle{\lim_{n \to \infty} \int_I f_n(x) \: dx}$ exists.
a) $(f_n(x))_{n=1}^{\infty}$ converges almost everywhere on $I$ to a limit function $f \in U(I)$.
b) $\displaystyle{\int_I f(x) \: dx = \int_I \lim_{n \to \infty} f_n(x) \: dx = \lim_{n \to \infty} \int_I f_n(x) \: dx}$.
Levi's Monotone Convergence Theorem for Upper Functions 1) $(f_n(x))_{n=1}^{\infty}$ is a sequence of upper functions that is increasing almost everywhere on $I$.
2) $\displaystyle{\lim_{n \to \infty} \int_I f_n(x) \: dx}$ exists.
a) $(f_n(x))_{n=1}^{\infty}$ converges almost everywhere on $I$ to a limit function $f \in U(I)$.
b) $\displaystyle{\int_I f(x) \: dx = \int_I \lim_{n \to \infty} f_n(x) \: dx = \lim_{n \to \infty} \int_I f_n(x) \: dx}$.
Levi's Monotone Convergence Theorem for Lebesgue Integrable Functions 1) $(f_n(x))_{n=1}^{\infty}$ is a sequence of Lebesgue integrable functions that is increasing almost everywhere on $I$.
2) $\displaystyle{\lim_{n \to \infty} \int_I f_n(x) \: dx}$ exists.
a) $(f_n(x))_{n=1}^{\infty}$ converges almost everywhere on $I$ to a limit function $f \in L(I)$.
b) $\displaystyle{\int_I f(x) \: dx = \int_I \lim_{n \to \infty} f_n(x) \: dx = \lim_{n \to \infty} \int_I f_n(x) \: dx}$.
Lebesgue's Dominated Convergence Theorem for Series 1) $(f_n(x))_{n=1}^{\infty}$ is a sequence of Lebesgue integrable functions.
2) Each $f_n$ is nonnegative.
3) $\displaystyle{\sum_{n=1}^{\infty} \int_I f_n(x) \: dx}$ converges almost everywhere on $I$.
a) $\displaystyle{\sum_{n=1}^{\infty} f_n(x)}$ converges almost everywhere on $I$ to a limit function $f \in L(I)$.
b) $\displaystyle{\int_I f(x) \: dx = \int_I \sum_{n=1}^{\infty} f_n(x) \: dx = \sum_{n=1}^{\infty} \int_I f_n(x) \: dx}$.
Fatou's Lemma 1) $(f_n(x))_{n=1}^{\infty}$ is a sequence of Lebesgue integrable functions that are nonnegative.
2) $(f_n(x))_{n=1}^{\infty}$ converges almost everywhere on $I$ to a limit function $f$.
3) There exists an $M \in \mathbb{R}$, $M > 0$ such that $\int_I f_n(x) \: dx \leq M$ for all $n \in \mathbb{N}$.
a) $f \in L(I)$.
b) $\displaystyle{\int_I f(x) \: dx \leq M}$.
Lebesgue's Dominated Convergence Theorem 1) $(f_n(x))_{n=1}^{\infty}$ is a sequence of Lebesegue integrable functions on $I$.
2) $(f_n(x))_{n=1}^{\infty}$ converges almost everywhere on $I$ to a limit function $f$.
3) There exists a nonnegative function $g \in L(I)$ such that $\mid f_n(x) \mid \leq g(x)$ almost everywhere on $I$ for all $n \in \mathbb{N}$.
a) $f \in L(I)$.
b) $\displaystyle{\int_I f(x) \: dx = \int_I \lim_{n \to \infty} f_n(x) \: dx = \lim_{n \to \infty} \int_I f_n(x) \: dx}$.
Lebesgue's Dominated Convergence Theorem for Series 1) $(f_n(x))_{n=1}^{\infty}$ is a sequence of Lebesgue integrable functions on $I$.
2) Each $f_n$ is nonnegative on $I$.
3) $\displaystyle{\sum_{n=1}^{\infty} f_n(x)}$ converges almost everywhere on $I$ to a limit function $f$.
4) There exists a nonnegative function $g \in L(I)$ such that $\mid f(x) \mid \leq g(x)$ almost everywhere on $I$.
a) $f \in L(I)$.
b) $\displaystyle{\int_I f(x) \: dx = \int_I \sum_{n=1}^{\infty} f_n(x) \: dx = \sum_{n=1}^{\infty} \int_I f_n(x) \: dx}$.
Lebesgue's Bounded Interval Convergence Theorem 1) $(f_n(x))_{n=1}^{\infty}$ is a sequence of Lebesgue integrable functions on $I$.
2) $(f_n(x))_{n=1}^{\infty}$ converges almost everywhere on $I$ to a limit function $f$.
3) There exists an $M \in \mathbb{R}$, $M > 0$ such that $\mid f_n(x) \mid \leq M$ almost everywhere on $I$ for all $n \in \mathbb{N}$.
a) $f \in L(I)$.
b) $\displaystyle{\int_I f(x) \: dx = \int_I \lim_{n \to \infty} f_n(x) \: dx = \lim_{n \to \infty} \int_I f_n(x) \: dx}$.
Test for Lebesgue Integrability 1) $(f_n(x))_{n=1}^{\infty}$ is a sequence of Lebesgue integrable functions on $I$.
2) $(f_n(x))_{n=1}^{\infty}$ converges almost everywhere on $I$ to a limit function $f$.
3) There exists a nonnegative function $g \in L(I)$ such that $\mid f(x) \mid \leq g(x)$ almost everywhere on $I$.
a) $f \in L(I)$.