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More in general, as we shall see in the next sections, if the boundary [partial derivative] A of a J-dimensional Borel set A [member of] [R.
Definition 2 (The class O) Let O be the class of Borel sets A of [R.
d-1] ([partial derivative]A); then it follows that this class of sets contains all Borel sets with (d-1)-rectifiable boundary (and so finite unions of sets with positive reach or with Lipschitz boundary, in particular).
there exists a constant C > 0 such that for any Borel set b of R, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denotes the norm in the Banach space B*.
For n [member of] Nand a Borel set b in R, put [[mu].
The next question we turn to is whether the distal flows form a Borel set.
This has the immediate corollaries that the distal flows of a given bounded rank form a Borel set and that the distal functions of a bounded rank form a Borel set.
k]) is a collection of Borel sets indexed by [Mathematical Expression Omitted] then "natural number quantification", which corresponds to countable unions and intersection, over this collection always yields a Borel set.
Namely, if [Mathematical Expression Omitted] for some Borel set E [subset] [C.
n-1](f(E)) = 0 for all Borel sets E [subset] [Delta][B.
The text covers sets and functions, Lebesgue measure, measurable sets, Borel sets
, sets of measure zero, the integral, measurable functions, convergence theorems, convergence in mean, Fourier theory, calculus, change of variables, differentiation of integrals, integration of derivatives, and general measures.