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Could you give me an example about:

"a collection of functions that is uniformly integrable but each (or some) function in the collection is not integrable."

This sounds counterintuitive? However according to Royden's "Real analysis":

  • Definition of "uniform integrability": A family $\mathcal F$ of measurable functions on $E$ is said to be uniformly integrable over $E$ if for every $\epsilon >0$ there is a $\delta >0$ such that for each $f \in \mathcal F$, if $A\subset E$ is measurable and $m(A)<\delta$, then $\int_A|f|<\epsilon$.

  • Property of integrability: When $m(E)<\infty$, $f$ is (Lebesgue) integrable over $E$ if and only if for every $\epsilon >0$ there is a $\delta >0$ such that if $A\subset E$ is measurable and $m(A)<\delta$, then $\int_A|f|<\epsilon$

When $m(E)=\infty$ then there is only one direction $\Rightarrow$ in the proposition about integrability right above. So I think my desired counter example should be about a collection of functions uniformly integrable on a set of infinite measure.

Thank you.

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    $\begingroup$ I doubt this is really what Royden means, but in any case the first definition is absurd. We can amuse ourselves by applying it to the "family of functions" consisting of the singleton $f(x)=1$ on $\mathbb{R}$ and conclude that this function is"uniformly integrable". $\endgroup$ – guest Jun 22 '15 at 0:50
  • $\begingroup$ I verbatim copy the definition of Royden (page 93), and I don't like this definition either since it induces the question above. I want a definition in which every individual function in a unifomrly integrable family of functions must be integrable. In your opinion, what is the definition of "uniform integrability" of a family of measurable functions. $\endgroup$ – Thang Jun 22 '15 at 0:55
  • $\begingroup$ I do not think it is a matter of opinion. A family of integrable functions is called uniformly integrable if the condition you stated holds. Note the prior requirement that the functions are integrable in the first place. The definition you wrote down is perfectly fine as long as separate integrability holds. Without separate integrability, the definition you stated essentially defines uniform local integrability. $\endgroup$ – guest Jun 22 '15 at 1:20
  • $\begingroup$ I can't find any line in Royden's textbook saying that each individual must be integrable before their collection is uniformly integrable. The gap lies in that when m(E) is infinity. $\endgroup$ – Thang Jun 22 '15 at 1:47
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I see your point, and guest's comment provides an immediate example ($\mathbb{R}\to\mathbb{R},x\mapsto1$) for a nonintegrable but uniformly integrable function.

I believe the reason why Royden & Fitzpatrick defines uniform integrability for a collection of measurable functions, instead of integrable functions, is to emphasize the analogy of this concept and equicontinuity, at least at the formal level:

  1. Let $\mathcal{F}$ be a collection of measurable functions from $E\subseteq\mathbb{R}$ to $\mathbb{R}$. Then

    • If $\mathcal{F}\subseteq L^1$, then $\forall f\in\mathcal{F},\forall\varepsilon>0,\exists\delta>0,\forall A\in\mathcal{M}:$

$$A\subseteq E, m(A)<\delta\implies \int_A |f|<\varepsilon.$$

  • $\mathcal{F}$ is uniformly integrable (or equiintegrable, or uniformly absolutely continuous) if $\forall\varepsilon>0,\exists\delta>0,\forall f\in\mathcal{F},\forall A\in\mathcal{M}:$

$$A\subseteq E, m(A)<\delta\implies \int_A |f|<\varepsilon.$$

  1. Let $(X,d)$ be a metric space, $x_0\in X$, $\mathcal{F}\subseteq\mathbb{R}^X$. Then
    • $\mathcal{F}$ is equicontinuous at $x_0$ if $\forall\varepsilon>0,\exists\delta>0,\forall f\in\mathcal{F},\forall x\in X:$

$$d(x,x_0)<\delta\implies |f(x)-f(x_0)|<\varepsilon.$$

Just as equicontinuity is generally not defined exclusively for continuous functions, Royden & Fitzpatrick chooses not to define uniform integrability exclusively for integrable functions.

Though admittedly this analogy is rather weak, and perhaps as a remedy to this all of the results in the corresponding section in the book relates either to collections of integrable functions, or collections of measurable functions with finite-measured domains (e.g. the Vitali Convergence Theorem).

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