Theorem of monotonic convergence

The theorem of monotonous convergence , also called Beppo Levi's theorem (after Beppo Levi ), is an important theorem from measurement and integration theory, a branch of mathematics. It makes a statement about the conditions under which integration and limit value formation can be interchanged.

Mathematical formulation

Be a measure space . If there is a sequence of non-negative, measurable functions that converges μ- almost everywhere monotonically increasing to a measurable function , then the following applies ${\displaystyle (\Omega ,{\mathcal {S}},\mu )}$ ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle f_{n}\colon \Omega \to [0,\infty ]}$ ${\displaystyle f\colon \Omega \to [0,\infty ]}$

${\displaystyle \int _{\Omega }f\ \mathrm {d} \mu =\lim _{n\to \infty }\int _{\Omega }f_{n}\ \mathrm {d} \mu .}$

Variant for falling episodes

If there is a function sequence of non-negative, measurable functions with which converges μ-almost everywhere monotonically decreasing to a measurable function , then the same applies ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle f_{n}\colon \Omega \to [0,\infty ]}$ ${\displaystyle \int _{\Omega }f_{1}\ \mathrm {d} \mu <\infty }$ ${\displaystyle f\colon \Omega \to [0,\infty ]}$

${\displaystyle \int _{\Omega }f\ \mathrm {d} \mu =\lim _{n\to \infty }\int _{\Omega }f_{n}\ \mathrm {d} \mu .}$

Proof idea

The fact that the right side is less than or equal to the left follows from the monotony of the integral. The other direction is decisive for the proof: This can be shown first for simple functions and from there transferred to general measurable functions.

Probability-theoretical formulation

Let be a probability space and a nonnegative, almost certainly monotonically growing sequence of random variables , then the following applies to their expectation values ${\displaystyle (\Omega ,{\mathcal {A}},P)}$ ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$

${\displaystyle \lim _{n\to \infty }\operatorname {E} (X_{n})=\operatorname {E} (\lim _{n\to \infty }X_{n})}$.

An analogous statement also applies to conditional expectation values : If a partial algebra is and integrable, then it is almost certain ${\displaystyle {\mathcal {G}}\subset {\mathcal {A}}}$ ${\displaystyle \sigma }$ ${\displaystyle \lim _{n\to \infty }X_{n}}$

${\displaystyle \lim _{n\to \infty }\operatorname {E} (X_{n}\mid {\mathcal {G}})=\operatorname {E} (\lim _{n\to \infty }X_{n}\mid {\mathcal {G}}).}$

Application of the theorem to series of functions

Be a measure space again. For every sequence of non-negative, measurable functions, the following applies ${\displaystyle (\Omega ,{\mathcal {S}},\mu )}$ ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle f_{n}\colon \Omega \to [0,\infty ]}$

${\displaystyle \int _{\Omega }\sum _{n=1}^{\infty }f_{n}\ \mathrm {d} \mu =\sum _{n=1}^{\infty }\int _{\Omega }f_{n}\ \mathrm {d} \mu .}$

This follows by applying the theorem to the sequence of partial sums. Since they are nonnegative, it is increasing monotonically. ${\displaystyle \textstyle s_{N}=\sum _{n=1}^{N}f_{n}}$ ${\displaystyle f_{n}}$ ${\displaystyle (s_{N})_{N\in \mathbb {N} }}$

See also

• Theorem of majorized convergence
• Lemma of Fatou

literature

• Elliott H. Lieb , Michael Loss : Analysis (= Graduate Studies in Mathematics. Vol. 14). 2nd edition. American Mathematical Society, Providence, RI 2001, ISBN 0-8218-2783-9 .

Individual evidence

1. ↑ Albrecht Irle : Probability Theory and Statistics: Basics - Results - Applications . 1st edition. Vieweg + Teubner, 2001, ISBN 978-3-519-02395-1 . Pages 116 to 118