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Let $f_k$ be a sequence of measurable functions on $\mathbb{R}^k$ where $k > 1$. (Let us be generous and also assume that $f_k$ is locally integrable.) Does anyone know what the phrase

uniform equicontinuity of the indefinite integrals $\int f_k(x) \mathrm{d}x$

means?

(For that matter, what is an indefinite integral on $\mathbb{R}^k$? The best I can figure is that it refers to the signed measure $Q \mapsto \int_{Q} f_k(x) \mathrm{d}x$, but I am not entirely sure how to interpret equicontinuity in this context.)

For what it is worth, the quote comes from Shatah and Struwe, Geometric Wave Equations, p.67 in the proof of Segal's theorem. I'll include the full context below.


The relevant setting: $f:\mathbb{R}\to\mathbb{R}$ is a locally Lipschitz function with $f(0) = 0$. We define $f_k:\mathbb{R}\to\mathbb{R}$ to be equal to $f$ on $[-k,k]$, and set $f_k(x) = f(k)$ if $x > k$ and similarly $f_k(x) = f(-k)$ if $x < -k$ (so it is a globally Lipschitz truncation of $f$).

Suppose we are now giving a sequence of functions $u_k \in L^2(I\times \mathbb{R}^m)$ where $I$ is a closed interval. We are given that $u_k \to u$ strongly in $L^2_{\text{loc}}$ and that $u_k \to u$ almost everywhere. We also assume that $u_k$ is uniformly bounded in $L^2(I\times\mathbb{R}^m)$. The claim is that knowing

$$ \int_I \int_{\mathbb{R}^m} |u_k f_k\circ u_k| \mathrm{d}x \leq \int_I\int_{\mathbb{R}^m} (u_k f_k\circ u_k + u_k^2) \mathrm{d}x + \int_I\int_{\mathbb{R}^m} u_k^2 \mathrm{d}x \leq C $$

and

$$ \int_I \int_{\mathbb{R}^m} |u f\circ u| \mathrm{d}x \leq \liminf_{k\to\infty} \int_I\int_{\mathbb{R}^m} (u_k f_k\circ u_k + u_k^2) \mathrm{d}x + \int_I\int_{\mathbb{R}^m} u^2 \mathrm{d}x \leq C $$

we can derive that the family of indefinite integrals $\int f_k\circ u_k \mathrm{d}x$ (no, no typos, it is not multiplied by $u_k$) is uniformly equicontinuous, and from this result we can get convergence of $f_k(u_k)\to f(u)$ in $L^1_{\text{loc}}$.

Now, from the final conclusion it appears that one may want to derive the conclusion using something like Vitali's theorem, which would mean that perhaps the authors intended the condition to be uniform integrability. Is that a reasonable interpretation?

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That's what I would think. Just check that it works with this interpretation (it certainly does in the passage you quoted but they might have used this expression in other parts of the text too) and forget about it :). Equicontinuity terminology for set functions starts making sense if you equip the set of measurable sets with the metric that is the measure of the symmetric difference. Then it agrees with the concept of uniform integrability in the setting under consideration. –  fedja Jun 1 '12 at 16:50
    
@fedja: if you put that as an answer, I will accept it (since it does answer the question I asked after all). Thanks. –  Willie Wong Jun 4 '12 at 8:14

1 Answer 1

up vote 1 down vote accepted

(Copying fedja's comment to get this off the unanswered list.)

That's what I would think. Just check that it works with this interpretation (it certainly does in the passage you quoted but they might have used this expression in other parts of the text too) and forget about it :). Equicontinuity terminology for set functions starts making sense if you equip the set of measurable sets with the metric that is the measure of the symmetric difference. Then it agrees with the concept of uniform integrability in the setting under consideration.

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