PlanetMath: proof of Egorov's theorem

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proof of Egorov's theorem

(Proof)

Let $E_{i,j} = \{x\in E: \vert f_j(x) - f(x)\vert< 1/i\}.$ Since $f_n\to f$ almost everywhere, there is a set with $\mu(S)=0$ such that, given $i\in \mathbb{N}$ and $x\in E-S$ , there is $m\in \mathbb{N}$ such that implies $\vert f_j(x)-f(x)\vert<1/i$ . This can be expressed by

$\displaystyle E-S\subset \bigcup _{m\in \mathbb{N}} \bigcap _{j>m} E_{i,j},$

or, in other words,

$\displaystyle \bigcap _{m\in \mathbb{N}}\bigcup _{j>m} (E-E_{i,j})\subset S.$

Since $\{\bigcup _{j>m} (E-E_{i,j})\}_{m\in \mathbb{N}}$ is a decreasing nested sequence of sets, each of which has finite measure, and such that its intersection has measure 0, by continuity from above we know that

$\displaystyle \mu(\bigcup _{j>m}(E-E_{i,j}))\xrightarrow[m\to \infty]{} 0.$

Therefore, for each $i\in \mathbb{N}$ , we can choose

such that

$\displaystyle \mu(\bigcup _{j>m_i}(E-E_{i,j})) < \frac{\delta}{2^i}.$

Let

$\displaystyle E_\delta = \bigcup _{i\in \mathbb{N}}\bigcup _{j>m_i}(E-E_{i,j}).$

Then

$\displaystyle \mu(E_\delta)\leq \sum_{i=1}^\infty \mu(\bigcup _{j>m_i}(E-E_{i,j})) < \sum_{i=1}^\infty \frac{\delta}{2^i} = \delta.$

We claim that $f_n\to f$ uniformly on $E-E_\delta$ . In fact, given $\varepsilon>0$ , choose

such that $1/n<\varepsilon$ . If $x\in E-E_\delta$ , we have

$\displaystyle x\in\bigcap _{i\in \mathbb{N}}\bigcap _{j>m_i}E_{i,j},$

which in particular implies that, if

, $x\in E_{n,j}$ ; that is, $\vert f_j(x) - f(x)\vert< 1/n < \varepsilon$ . Hence, for each $\varepsilon>0$ there is

(which is given by

above) such that

implies $\vert f_j(x)-f(x)\vert<\varepsilon$ for each $x\in E-E_\delta$ , as required. This completes the proof.

"proof of Egorov's theorem" is owned by Koro.

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Cross-references: intersection, measure, finite, sequence, decreasing, implies, almost everywhere

This is version 4 of proof of Egorov's theorem, born on 2003-07-27, modified 2006-07-27.
Object id is 4518, canonical name is ProofOfEgorovsTheorem.
Accessed 4663 times total.

Classification:

AMS MSC:

28A20 (Measure and integration :: Classical measure theory :: Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence)

Pending Errata and Addenda

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