A Generalization of Lusin's Theorem

In this note we characterize cr-finite Riesz measures that allow one to approximate measurable functions by continuous functions in the sense of Lusin's theorem. We call such measures Lusin measures and show that not all cr-finite measures are Lusin measures. It is shown that if a topological space X is either normal or countably paracompact, then every measure on A' is a Lusin measure. A counterexample is given to show that these sufficient conditions are not necessary. 0. Introduction and results. We term a positive measure p a Riesz measure if it has the properties stated in the Riesz representation theorem, i.e. if it is a positive measure on a locally compact Hausdorff space X with the additional properties: (a) ft is defined on a cr-algebra M which contains all Borel sets in X. (b) piK) < oo for every compact set K C X. (c) For every E £ M, we have piE) = inf ift(V)|F C V, V is open!. (d) For every E £ M such that either piE) < oo or E is open we have piE) = sup{/x(K)|r< C F, K is compact!. Together, (a) and (c) state that p is an outer regular Borel measure. Inner regularity is required by (d) only on open sets and sets of finite measure. By a space, we will always mean a locally compact Hausdorff space, and by a measure, we will mean a Riesz measure. The following is a standard result due to Lusin. Lusin's theorem. Suppose f is a complex measurable function on a space X, piX) < oo, and e > 0. Then there exists g, a continuous function on X, such that p{x\f(x) 4 gix)\ < e. We will call p a Lusin measure if ft is a cr-finite Riesz measure on a locally compact Hausdorff space for which the conclusion of Lusin's theorem is true. A locally compact Hausdorff space X is a Lusin space if every afinite Riesz measure ft on X is a Lusin measure. In this language, Lusin's theorem says that a finite Riesz measure is a Lusin measure. Lebesgue measure on R" can easily be shown to be a Lusin measure. R" is cr-compact and Lebesgue measure is cr-finite. Therefore it is natural to ask if all ocompact topological spaces are Lusin spaces and if all o^finite measures are Lusin measures. It is the purpose of this note to answer these and similar questions. Received by the editors May 1, 1974 and, in revised form, July 10, 1974. AMS (MOS) subject classifications (1970). Primary 26A15, 28A10; Secondary 54G20. Copyright © 1975, American Mathematical Society 327 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use