|
In mathematics, the Riesz–Markov–Kakutani representation theorem relates linear functionals on spaces of continuous functions on a locally compact space to measures. The theorem is named for who introduced it for continuous functions on the unit interval, who extended the result to some non-compact spaces, and who extended the result to compact Hausdorff spaces. There are many closely related variations of the theorem, as the linear functionals can be complex, real, or positive, the space they are defined on may be the unit interval or a compact space or a locally compact space, the continuous functions may be vanishing at infinity or have compact support, and the measures can be Baire measures or regular Borel measures or Radon measures or signed measures or complex measures. == The representation theorem for positive linear functionals on ''Cc''(''X'') == The following theorem represents positive linear functionals on ''Cc''(''X''), the space of continuous compactly supported complex-valued functions on a locally compact Hausdorff space ''X''. The Borel sets in the following statement refer to the σ-algebra generated by the ''open'' sets. A non-negative countably additive Borel measure μ on a locally compact Hausdorff space ''X'' is regular if and only if * μ(''K'') < ∞ for every compact ''K''; * For every Borel set ''E'', :: * The relation :: holds whenever ''E'' is open or when ''E'' is Borel and ''μ''(''E'') < ∞ . Theorem. Let ''X'' be a locally compact Hausdorff space. For any positive linear functional on ''C''''c''(''X''), there is a unique regular Borel measure μ on ''X'' such that : for all ''f'' in Cc(''X''). One approach to measure theory is to start with a Radon measure, defined as a positive linear functional on ''C''(''X''). This is the way adopted by Bourbaki; it does of course assume that ''X'' starts life as a topological space, rather than simply as a set. For locally compact spaces an integration theory is then recovered. Without the condition of regularity the Borel measure need not be unique. For example, let ''X'' be the set of ordinals at most equal to the first uncountable ordinal Ω, with the topology generated by "open intervals". The linear functional taking a continuous function to its value at Ω corresponds to the regular Borel measure with a point mass at Ω. However it also corresponds to the (non-regular) Borel measure that assigns measure 1 to any measurable subset of the space of ordinals less than Ω that is closed and unbounded, and assigns measure 0 to other measurable subsets. Historical remark: In its original form by F. Riesz (1909) the theorem states that every continuous linear functional ''A''() over the space ''C''((1 )) of continuous functions in the interval () can be represented in the form : where ''α''(''x'') is a function of bounded variation on the interval (1 ), and the integral is a Riemann–Stieltjes integral. Since there is a one-to-one correspondence between Borel regular measures in the interval and functions of bounded variation (that assigns to each function of bounded variation the corresponding Lebesgue–Stieltjes measure, and the integral with respect to the Lebesgue–Stieltjes measure agrees with the Riemann–Stieltjes integral for continuous functions ), the above stated theorem generalizes the original statement of F. Riesz. (See Gray(1984), for a historical discussion). 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「Riesz–Markov–Kakutani representation theorem」の詳細全文を読む スポンサード リンク
|