Egoroff's Theorem and the Distribution of Standard Points in a Nonstandard Model1

We study the relationship between the Loeb measure °(*fi) of a set £ and the /¿-measure of the set S(E) = [x\*x G E) of standard points in E. If E is in the a-algebra generated by the standard sets, then °(*f0(£) — piS(Ef). This is used to give a short nonstandard proof of Egoroffs Theorem. If £ is an internal, * measurable set, then in general there is no relationship between the measures of S(£) and E. However, if *X is an ultrapower constructed using a minimal ultrafilter on u, then *p(£) «¡ 0 implies that S(E) is a u-null set. If, in addition, y. is a Borel measure on a compact metric space and £ is a Loeb measurable set, then M(5(£)) <°(*f)(£) < ß(S(E)) where p and fi are the inner and outer measures for p. The work in this paper was originally stimulated by the search for an illuminating nonstandard proof of Egoroffs Theorem. Despite the importance of such a proof it has been surprisingly elusive (see, for example, [8] or [11]). §1 of this paper presents a short, natural proof of Egoroffs Theorem using a result from §11 on the distribution of standard points in a nonstandard model. The work in §11 is of independent interest. Throughout this paper (A', &, y) will denote a (standard) positive measure space with y(X) finite; 91L will denote a standard higher order model of X along with the real numbers, R; and *<3H will denote a proper nonstandard extension of R and /: X -» 7? is also a measurable function. Egoroffs Theorem [3] states 1.1 Egoroff's Theorem. 7/ S„ —* S pointwise almost everywhere then Sor every e > 0 there is a set A G <$ such that y(A) < e and/„ -»/ uniSormly on X \ A. Received by the editors July 30, 1979 and, in revised form, February 5, 1980; presented to the Society, January 4, 1980 at the Special Session on Nonstandard Analysis, San Antonio, Texas. 1980 Mathematics Subject Classification. Primary 03H05. 'This work was partially supported by grants from the National Science Foundation. © 1981 American Mathematical Society 0002-9939/81/0000-012 5/$02.7 5 455 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 456 C. W. HENSON AND FRANK WATTENBERG If {S„} satisfies the conclusion of Egoroffs Theorem we will say {/„} converges to / nearly uniSormly. Note that Egoroffs Theorem is false without the assumption that y(X) is finite. The following characterization of nearly uniform convergence is essentially due to Robinson [8]. 1.2 Definition. Suppose/,/,,/2, ... are standard measurable functions X ^> R, and x G *X. Then x is said to be a point of intrinsic nonuniformity if there is an infinite integer v such that/,,(*) »*/(■*)• Let E denote the set of points of intrinsic nonuniformity. (Note: E is usually external.) 1.3 Definition. Suppose A is a (possibly external) subset of *X. A is said to have S-measure zero if for every standard e > 0 there is a standard set B G IS such that A G*B and y(B) < e. — r-v / 1.4 Proposition. Using the notation oj DeSinition 1.2, the Sollowing are equivalent. (i) {/„} converges to S nearly uniSormly. (ii) E has S-measure zero. Proof. The proof is completely straightforward using the well-known fact that S„-*S uniformly on a set S if and only if for every p G*S and every infinite v, JÁP) ~ */(/>) [8, Theorem 4.6.1]. We need one more definition before proving Egoroffs Theorem. 1.5 Definition. Suppose A is a (possibly external) subset of *X. Let S(A) denote the set of all standard points in A. That is, S(A) = {x G X\*x G A). Note S(A) is just the standard part of A with respect to the discrete topology on X. 1.6 Proof of Egoroff's Theorem. Suppose/, —>/pointwise almost everywhere. Hence there is a set A G 6J such that y(A) = 0 and/„ -»/pointwise on X \ A. Let E denote the set of points of intrinsic nonuniformity. Then S(E) G A. Thus S(E) has measure zero and by II.3 E has S-measure zero, completing the proof by 1.4. II. The distribution of standard points in *X. The purpose of this section is to study the relationship between the measure (in a sense to be defined below) of a set E G*X and the standard measure of S(E). Intuitively, the standard points are evenly distributed in *X and one might, therefore, expect the measures of E and S(E) to be infinitely close for a reasonable class of sets E. II. 1 Definition. Let éE be the (external) algebra, & = {*A\A G <$}, and let S be the (external) a-algebra generated by é£. Using the Loeb-Carathéodory extension process there is an (external) real-valued a-additive measure °(*y): S -* [0, oo) [5], see also [8, §5.1], called S-measure. Notice °(*y)(A) = 0 if and only if A has S-measure zero in the sense of 1.3. II.2 Theorem. Suppose E G S. Then S(E) G ^ and °(*y)(E) = y(S(E)). Proof. First, let 3", = {E G § |S(7Í) G <$}. % is a a-algebra since S(*X \ A) = X \ S(A), S(AX n A2) = S(AX) n S(A2) and S(U"_i4.) = U ?-\S(AJ. Hence S = 5,. Now we have two finite measures defined on §, yx(E) =°(*y)(E) and y^(E) = y(S(E)). By the uniqueness part of the Caratheodory Extension Theorem, we have yx = y? completing the proof. Notice the importance here that y(X) is finite. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use DISTRIBUTION OF STANDARD POINTS 457 11.3 Example. Let E be as in 1.6, the set of points of intrinsic nonuniformity for/ and (/,) where/, -»/almost everywhere. Then £ G S and °(*ft)(7i) = y(S(E)) = 0. Proof. Let^ = {x G X\3r > k\Sr(x) *f(x)\ > I/«}Claim: