Pratt's theorem

The set of Pratt is a mathematical set of measure theory , a generalization of the set of the dominated convergence and a variation of the measure-theoretic Einschnürungssatzes corresponds. The sentence clearly states that if a sequence of functions is almost everywhere between two further sequences of functions and these converge and allow limit value formation and integration to be interchanged, the function series in brackets also allows limit value formation and integration to be interchanged. The theorem was proved by John W. Pratt in 1960 .

statement

A measurement space and a sequence of measurable functions are given ${\displaystyle (X,{\mathcal {A}},\mu )}$

${\displaystyle (f_{n}\colon X\to \mathbb {R} )_{n\in \mathbb {N} }}$

from the local-to-measure against converges. In addition, let the set be σ-finite . ${\displaystyle {\mathcal {L}}^{1}(\mu )}$ ${\displaystyle f}$ ${\displaystyle \{f\neq 0\}}$

Now exist in , for which: ${\displaystyle g,h\colon X\to \mathbb {R} ,(g_{n}\colon X\to \mathbb {R} )_{n\in \mathbb {N} },(h_{n}\colon X\to \mathbb {R} )_{n\in \mathbb {N} }}$ ${\displaystyle {\mathcal {L}}^{1}(\mu )}$

1. ${\displaystyle (g_{n})_{n\in \mathbb {N} }}$converges locally to measure and locally converges to measure . ${\displaystyle g}$ ${\displaystyle (h_{n})_{n\in \mathbb {N} }}$ ${\displaystyle h}$
2. This applies to everyone - almost everywhere ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle \mu }$

   ${\displaystyle g_{n}\leq f_{n}\leq h_{n}}$.

3. It is

   ${\displaystyle \lim _{n\to \infty }\int _{X}g_{n}\mathrm {d} \mu =\int _{X}g\mathrm {d} \mu {\text{ und }}\lim _{n\to \infty }\int _{X}h_{n}\mathrm {d} \mu =\int _{X}h\mathrm {d} \mu }$.

Then is also made and it is ${\displaystyle f}$ ${\displaystyle {\mathcal {L}}^{1}(\mu )}$

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

Example: majorized convergence

A modification of the principle of majorized convergence follows directly from the sentence . If the conditions above in the definition of an integrable positive majorant of , ${\displaystyle m}$ ${\displaystyle f}$

it has already made and it is ${\displaystyle f}$ ${\displaystyle {\mathcal {L}}^{1}(\mu )}$

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

To do this, one sets as function sequences

${\displaystyle g_{n}=g=-m{\text{ und }}h_{n}=h=m}$.

Due to the constancy of the function sequences, the interchanging of limit value and integral is given and they can be integrated because they agree with the integrable majorant. In addition, the function sequences also converge locally to measure, since they are constant. It is the same as with the principle of majorized convergence or almost everywhere. Thus all three conditions are fulfilled and Pratt's theorem delivers the statement. ${\displaystyle g_{n},g,h_{n},h}$ ${\displaystyle |f_{n}|\leq m}$ ${\displaystyle g_{n}\leq f_{n}\leq h_{n}}$

In contrast to the theorem of majorized convergence, the statement applies here if the locally converged to measure and not, as originally required for the majorized convergence, point by point almost everywhere . ${\displaystyle f_{n}}$ ${\displaystyle f}$

literature

• Jürgen Elstrodt : Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , doi : 10.1007 / 978-3-540-89728-6 .