Lemma of Fatou

The lemma of Fatou (after Pierre Fatou ) allows in mathematics to estimate the Lebesgue integral of the Limes inferior of a function sequence through the Limes inferior of the sequence of the associated Lebesgue integrals upwards. It thus provides a statement about the interchangeability of limit value processes .

Mathematical formulation

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

${\displaystyle \int _{S}\liminf _{n\rightarrow \infty }f_{n}\ \mathrm {d} \mu \leq \liminf _{n\rightarrow \infty }\int _{S}f_{n}\ \mathrm {d} \mu ,}$

whereby on the left side the Limes inferior of the sequence is to be understood point by point . ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$

Analogously, this set also for the Limes superior, unless a non-negative integrable function with are: ${\displaystyle g}$ ${\displaystyle f_{n}\leq g}$

${\displaystyle \int _{S}\limsup _{n\rightarrow \infty }f_{n}\ \mathrm {d} \mu \geq \limsup _{n\rightarrow \infty }\int _{S}f_{n}\ \mathrm {d} \mu }$.

This can be summarized as the rule of thumb

${\displaystyle \int _{S}\liminf _{n\rightarrow \infty }f_{n}\ \mathrm {d} \mu \leq \liminf _{n\rightarrow \infty }\int _{S}f_{n}\ \mathrm {d} \mu \leq \limsup _{n\rightarrow \infty }\int _{S}f_{n}\ \mathrm {d} \mu \leq \int _{S}\limsup _{n\rightarrow \infty }f_{n}\ \mathrm {d} \mu }$.

Proof idea

In order to prove Fatou's lemma for the Limes inferior, one turns to the monotonically increasing sequence of functions

${\displaystyle g_{n}:=\inf _{k\geq n}f_{k}\nearrow \liminf _{n\rightarrow \infty }f_{n}}$

the theorem of monotonous convergence . With the resulting equation and the inequality based on the monotony of the integral

${\displaystyle \int _{S}\left(\inf _{k\geq n}f_{k}\right)\leq \int _{S}f_{n}}$

one obtains from the calculation rules for the Limes:

${\displaystyle \int _{S}\liminf _{n\rightarrow \infty }f_{n}=\lim _{n\rightarrow \infty }\int _{S}\left(\inf _{k\geq n}f_{k}\right)\leq \liminf _{n\rightarrow \infty }\int _{S}f_{n}}$.

For Fatou's lemma with limes superior analog can be moved, because, by assumption, is to be integrated, that is integrated. ${\displaystyle g_{1}=\sup _{k\geq 1}f_{k}\leq g}$ ${\displaystyle g}$ ${\displaystyle g_{1}}$

Examples of strict inequality

The base space is provided with Borel's σ-algebra and the Lebesgue measure . ${\displaystyle S}$

• Example for a probability space : Let be the unit interval. Define for all and , where denotes the indicator function of the interval . ${\displaystyle S=[0,1]}$ ${\displaystyle f_{n}(x)=n\mathbf {1} _{(0,{\frac {1}{n}})}(x)}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle x\in S}$ ${\displaystyle \mathbf {1} _{(0,{\frac {1}{n}})}}$ ${\displaystyle (0,{\tfrac {1}{n}})}$
• Example with uniform convergence : Let be the set of real numbers . Define for everyone and . (Note that there is no integrable majorante in this example and therefore the sup part of Fatou's lemma is not applicable.) ${\displaystyle S}$ ${\displaystyle f_{n}(x)={\tfrac {1}{n}}\mathbf {1} _{[0,n]}(x)}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle x\in S}$

Each has an integral one, ${\displaystyle f_{n}}$

${\displaystyle \int _{S}f_{n}\ \mathrm {d} \mu =1}$

therefore applies

${\displaystyle 1=\lim _{n\rightarrow \infty }\int _{S}f_{n}\ \mathrm {d} \mu =\liminf _{n\rightarrow \infty }\int _{S}f_{n}\ \mathrm {d} \mu =\limsup _{n\rightarrow \infty }\int _{S}f_{n}\ \mathrm {d} \mu }$

The sequence converges point by point to the null function ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle S}$

${\displaystyle 0=\lim _{n\rightarrow \infty }f_{n}=\liminf _{n\rightarrow \infty }f_{n}=\limsup _{n\rightarrow \infty }f_{n},}$

therefore the integral is also zero

${\displaystyle 0=\int _{S}\liminf _{n\rightarrow \infty }f_{n}\ \mathrm {d} \mu =\int _{S}\limsup _{n\rightarrow \infty }f_{n}\ \mathrm {d} \mu ,}$

therefore the strict inequalities apply here

${\displaystyle \int _{S}\liminf _{n\rightarrow \infty }f_{n}\ \mathrm {d} \mu <\liminf _{n\rightarrow \infty }\int _{S}f_{n}\ \mathrm {d} \mu ,}$

${\displaystyle \int _{S}\limsup _{n\rightarrow \infty }f_{n}\ \mathrm {d} \mu <\limsup _{n\rightarrow \infty }\int _{S}f_{n}\ \mathrm {d} \mu }$

Discussion of the requirements

The precondition of the nonnegativity of the individual functions cannot be dispensed with, as the following example shows: Let the half-open interval with Borel's σ-algebra and the Lebesgue measure. Define for everyone . The sequence converges on (even uniformly) to the null function (with integral 0), but each has an integral −1. thats why ${\displaystyle S}$ ${\displaystyle [0,\infty )}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle f_{n}(x):=-{\tfrac {1}{n}}{\mathfrak {1}}_{[0,n]}(x)}$ ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle S}$ ${\displaystyle f_{n}}$

${\displaystyle 0=\int _{S}\lim _{n\rightarrow \infty }f_{n}\ \mathrm {d} \mu >\lim _{n\rightarrow \infty }\int _{S}f_{n}\mathrm {d} \mu =-1}$.

See also

• Theorem of majorized convergence
• Theorem of monotonic convergence

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 .
• Walter Rudin : Analysis. German edition revised by Norbert Herrmann . 2nd, corrected edition. Oldenbourg, Munich et al. 2002, ISBN 3-486-25810-9 , p. 376: Chapter 11, sentence 11.31.