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In mathematics, particularly in the area of functional analysis and topological vector spaces, the vague topology is an example of the weak- * topology which arises in the study of measures on locally compact Hausdorff spaces. Let ''X'' be a locally compact Hausdorff space. Let ''M''(''X'') be the space of complex Radon measures on ''X'', and ''C''0(''X'') * denote the dual of ''C''0(''X''), the Banach space of complex continuous functions on ''X'' vanishing at infinity equipped with the uniform norm. By the Riesz representation theorem ''M''(''X'') is isometric to ''C''0(''X'') *. The isometry maps a measure ''μ'' to a linear functional : The vague topology is the weak- * topology on ''C''0(''X'') *. The corresponding topology on ''M''(''X'') induced by the isometry from ''C''0(''X'') * is also called the vague topology on ''M''(''X''). Thus, in particular, one may refer to vague convergence of measure ''μ''''n'' → ''μ''. One application of this is to probability theory: for example, the central limit theorem is essentially a statement that if ''μ''''n'' are the probability measures for certain sums of independent random variables, then ''μ''''n'' converge weakly (and then vaguely) to a normal distribution, i.e. the measure ''μ''''n'' is "approximately normal" for large ''n''. ==References== * . * G.B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed, John Wiley & Sons, Inc., 1999. 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「Vague topology」の詳細全文を読む スポンサード リンク
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