Page 159

		globally good — are locally terrible. As an example, let . Clearly, den(C) = 1 and, if p and f are such that , then aa( j p, f) = 1. Thus the asymptotic agreement between  j p and f is as good as possible, but contains arbitrarily large gaps. The following definition addresses this defect in Aex by introducing a stricter notion of density.

		7.15 Definition

		(a) The uniform density of A in intervals of (written: udenm (A)) is .

		(b) The uniform density of A (written: uden(A)) is udm(A).1

		(c) The asymptotic uniform agreement between A and B (written: aua(A, B)) is . The asymptotic uniform agreement between  a  and  b  (written: aua( a ,  b )) is .

		(d) The asymptotic uniform disagreement between A and B (written: aud(A, B)) is 1 -aua(A, B). The asymptotic uniform disagreement between  a  and  b  (written: aud( a ,  b )) is 1 - aua( a ,  b ).

		So, if , then for each there is an such that for every interval I of length at least , . In particular, if C is as above, then for each m, udm(C) = 0; hence uden(C) = 0.

		7.16 Definition Suppose a is a real in .

		(a) An M Uexa-identifies f (written: ) if and only if and .

		(b) .

		For example, if M Uex0-identifies f with p = M(f), then

		,

		thus p is a uniformly good approximation to f. The relationship of the Uex criteria to the Ex, Bc, and Aex criteria is easy to establish.

		1Note that for each A and m, , and hence by elementary real analysis, the limit udm(A) always exists.