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The definition of uniform integrability of a family of $L^1$ functions is:

If μ is a finite measure, a subset $K \subset L^1(\mu) $ is said to be uniformly integrable if $\lim_{c \to \infty} \sup_{X \in K} \int_{|X|\geq c} |X|\, d\mu = 0.$

It inspires the following attempt to coordinate convergence of each in a function family as approaching a domain limit point:

  1. Suppose $\{f_a\}_{a \in A}$ is a family of functions from $U \subseteq \mathbb{R} \to \mathbb{R}$, $x_0$ is a limit point of $U$ in $\mathbb{R}$, and $L \in \mathbb{R}$. Are there some existing concepts for cases similar to the one when $\lim_{x\to x_0} \sup_{a \in A} |f_a(x) -L| = 0$?
  2. Can such concepts be generalized for a family of mappings defined from a subset of a topological space to a metric space? By replacing absolute value with metric, I think the example in 1 can. Can the codomains be generalized to a uniform space or even a topological space?
  3. Would it be called "equi-convergence", or "uniform convergence" (bad name as it is already used for another purpose) in the same spirit of "uniform integrability"? This is because I notice the difference between the naming of "equi"-continuity of a family of mappings at a domain point, and the naming of "uniform" continuity of a mapping over its domain ("uniform" seems always suggest the existence of a uniform space).

    In the same spirit, do you think "uniform integrability" is really a bad name, and probably "equi-integrability" will be a better one?

Thanks and regards!

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    @DavideGiraudo: Tha$n$ks! "whe$n$ you have a family of random variables this limit is uniform", but this "uniform" is over the family of random variables, not over their domain. So I would think "equi" will be better, as it is over a family of mappings, such as in equicontinuity.2012-02-22

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