Summary of Convergence Theorems for Lebesgue Integration

# Summary of Convergence Theorems for Lebesgue Integration

We will now summarize the convergence theorems that we have looked at regarding Lebesgue integration.

For each of the convergence theorems below we assume that $E$ is a Lebesgue measurable set.

#### Convergence Theorems for Bounded Lebesgue Measurable Functions

The Pointwise Bounded Convergence Theorem for Uniformly Convergent Sequences of Functions |
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Conditions | 1. $m(E) < \infty$.2. $(f_n(x))_{n=1}^{\infty}$ is a sequence of Lebesgue measurable functions on $E$.3. $(f_n(x))_{n=1}^{\infty}$ is a set of bounded functions on $E$.4. $(f_n(x))_{n=1}^{\infty}$ converges uniformly to $f(x)$ on $E$. |
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Conclusions | 1. $\displaystyle{\lim_{n \to \infty} \int_E f_n = \int_E f}$. |

The Uniform Bounded Convergence Theorem for Pointwise Convergent Sequences of Functions |
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Conditions | 1. $m(E) < \infty$.2. $(f_n(x))_{n=1}^{\infty}$ is a sequence of Lebesgue measurable functions on $E$.3. $(f_n(x))_{n=1}^{\infty}$ is a set of functions that is uniformly bounded on $E$.4. $(f_n(x))_{n=1}^{\infty}$ converges pointwise to $f(x)$ on $E$. |
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Conclusions | 1. $\displaystyle{\lim_{n \to \infty} \int_E f_n = \int_E f}$. |

#### Convergence Theorems for Nonnegative Lebesgue Measurable Functions

Fatou's Lemma for Nonnegative Lebesgue Measurable Functions |
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Conditions | 1. $(f_n(x))_{n=1}^{\infty}$ is a sequence of Lebesgue measurable functions on $E$.2. $f_n(x) \geq 0$ for all $n \in \mathbb{N}$ and for all $x \in E$.3. $(f_n(x))_{n=1}^{\infty}$ converges pointwise to $f(x)$ on $E$. |
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Conclusions | $\displaystyle{\int_E f \leq \liminf_{n \to \infty} \int_E f_n}$. |

Lebesgue's Monotone Convergence Theorem |
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Conditions | 1. $(f_n(x))_{n=1}^{\infty}$ is a sequence of Lebesgue measurable functions on $E$.2. $f_n(x) \geq 0$ for all $n \in \mathbb{N}$ and for all $x \in E$.3. $0 \leq f_1(x) \leq f_2(x) \leq ... \leq f_n(x) \leq ... \leq f(x)$ for all $x \in E$.4. $(f_n(x))_{n=1}^{\infty}$ converges pointwise to $f(x)$ on $E$. |
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Conclusions | 1. $\displaystyle{\lim_{n \to \infty} \int_E f_n = \int_E f}$. |

Lebesgue's Monotone Convergence Theorem for Series |
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Conditions | 1. $(f_n(x))_{n=1}^{\infty}$ is a sequence of Lebesgue measurable functions on $E$.2. $f_n(x) \geq 0$ for all $n \in \mathbb{N}$ and for all $x \in E$.3. $\displaystyle{\sum_{n=1}^{\infty} f_n(x)}$ converges on $E$. |
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Conclusions | 1. $\displaystyle{\sum_{n=1}^{\infty} \int_E f_n = \int_E \sum_{n=1}^{\infty} f_n}$. |

#### Convergence Theorems for Lebesgue Measurable Functions

The Comparison Test for Lebesgue Integrability |
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Conditions | 1. $f$ is a Lebesgue measurable function on $E$.2. $g$ is Lebesgue integrable function on $E$.3. $g(x) \geq 0$ for all $x \in E$.4. $|f(x)| \leq g(x)$ almost everywhere on $E$. |
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Conclusions | 1. $f$ is Lebesgue integrable on $E$. |

The Lebesgue Dominated Convergence Theorem |
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Conditions | 1. $(f_n(x))_{n=1}^{\infty}$ is a sequence of Lebesgue measurable functions on $E$.2. There exists a nonnegative Lebesgue integrable function $g$ on $E$ with $|f_n(x)| \leq g(x)$ for all $n \in \mathbb{N}$ and for all $x \in E$.3. $(f_n(x))_{n=1}^{\infty}$ converges pointwise almost everywhere to $f(x)$ on $E$. |
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Conclusions | 1. $f$ is Lebesgue integrable on $E$.2. $\displaystyle{\lim_{n \to \infty} \int_E f_n = \int_E f}$. |

The Generalized Lebesgue Dominated Convergence Theorem |
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Conditions | 1. $(f_n(x))_{n=1}^{\infty}$ is a sequence of Lebesgue measurable functions on $E$.2. $(g_n(x))_{n=1}^{\infty}$ is a sequence of nonnegative Lebesgue measurable functions on $E$.3. $|f_n(x)| \leq g_n(x)$ for all $n \in \mathbb{N}$ and for all $x \in E$.4. $(f_n(x))_{n=1}^{\infty}$ converges pointwise almost everywhere to $f(x)$ on $E$.5. $(g_n(x))_{n=1}^{\infty}$ converges pointwise almost everywhere to $g(x)$ on $E$.6. $\displaystyle{\lim_{n \to \infty} \int_E g_n = \int_E g < \infty}$. |
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Conclusions: | 1. $f$ is Lebesgue integrable on $E$.2. $\displaystyle{\lim_{n \to \infty} \int_E f_n = \int_E f}$. |