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In mathematics, more specifically measure theory, there are various notions of the convergence of measures. For an intuitive general sense of what is meant by ''convergence in measure'', consider a sequence of measures μ''n'' on a space, sharing a common collection of measurable sets. Such a sequence might represent an attempt to construct 'better and better' approximations to a desired measure μ that is difficult to obtain directly. The meaning of 'better and better' is subject to all the usual caveats for taking limits; for any error tolerance ε > 0 we require there be ''N'' sufficiently large for ''n'' ≥ ''N'' to ensure the 'difference' between μ''n'' and μ is smaller than ε. Various notions of convergence specify precisely what the word 'difference' should mean in that description; these notions are not equivalent to one another, and vary in strength. Three of the most common notions of convergence are described below. ==Informal descriptions== This section attempts to provide a rough intuitive description of three notions of convergence, using terminology developed in calculus courses; this section is necessarily imprecise as well as inexact, and the reader should refer to the formal clarifications in subsequent sections. In particular, the descriptions here do not address the possibility that the measure of some sets could be infinite, or that the underlying space could exhibit pathological behavior, and additional technical assumptions are needed for some of the statements. The statements in this section are however all correct if is a sequence of probability measures on a Polish space. The various notions of convergence formalize the assertion that the 'average value' of each 'sufficiently nice' function should converge: : To formalize this requires a careful specification of the set of functions under consideration and how uniform the convergence should be. The notion of ''weak convergence'' requires this convergence to take place for every continuous bounded function . This notion treats convergence for different functions ''f'' independently of one another, ''i.e.'' different functions ''f'' may require different values of ''N'' ≤ ''n'' to be approximated equally well (thus, convergence is non-uniform in ). The notion of ''strong convergence'' formalizes the assertion that the measure of each measurable set should converge: : Again, no uniformity over the set is required. Intuitively, considering integrals of 'nice' functions, this notion provides more uniformity than weak convergence. As a matter of fact, when considering sequences of measures with uniformly bounded variation on a Polish space, strong convergence implies the convergence for any bounded measurable function . As before, this convergence is non-uniform in The notion of ''total variation convergence'' formalizes the assertion that the measure of all measurable sets should converge ''uniformly'', i.e. for every there exists ''N'' such that for every ''n > N'' and for every measurable set . As before, this implies convergence of integrals against bounded measurable functions, but this time convergence is uniform over all functions bounded by any fixed constant. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Convergence of measures」の詳細全文を読む スポンサード リンク
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