Scheffé's Lemma/Corollary

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Corollary to Scheffé's Lemma

Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $f_n$ be a sequence of $\mu$-integrable functions that convergence in measure to another $\mu$-integrable function $f$.

Then $f_n$ converges to $f$ in $L^1$ if and only if $\ds \int_X \size {f_n} \rd \mu$ converges to $\ds \int_X \size f \rd \mu$.

Proof

Although this article appears correct, it's inelegant. There has to be a better way of doing it. In particular: The proof of this is heavily dependent upon the structure of the proof of Scheffé's Lemma, and hence is not consistent with the structure of pages in $\mathsf{Pr} \infty \mathsf{fWiki}$ whereby the only thing we can base a proof on is a link to an actual result. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by redesigning it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Improve}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Let $f_n$ converges to $f$ in measure instead.

The proof of the first direction remains unchanged from the above.

In the other direction, suppose:

$\ds \int_X \size {f_n} \rd \mu \to \int_X \size f \rd \mu$

We wish to show again that:

$\ds \int_X \size {f - f_n} \rd \mu \to 0$.

Aiming for a contradiction, suppose this is false.

Since the integral is non-negative, we can find some $\epsilon > 0$ and an infinite subsequence $g_n$ of $f_n$ such that:

$\ds \int_X \size {f - g_n} \rd \mu \ge \epsilon$

Since $g_n$ still converges in measure to $f$, by Convergence in Measure Implies Convergence a.e. of Subsequence $g_n$ has a further subsequence $h_n$ that converges almost everywhere to $f$.

But we also have:

$\ds \int_X \size {h_n} \rd \mu \to \int_X \size f \rd \mu$

Hence from Scheffé's Lemma:

$\ds \int_X \size {f - h_n} \rd \mu \to 0$

This is a contradiction, as $h_n$ is a subsequence of $g_n$, hence $\ds \int_X \size {f - h_n} \rd \mu$ must remain larger than $\epsilon$.

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$\blacksquare$

Source of Name

This entry was named for Henry Scheffé.