For instance, a Besicovitch showed that there are Besicovitch sets of measure zero.
2.
*" f " carries closed sets of measure zero onto closed sets of measure zero.
3.
*" f " carries closed sets of measure zero onto closed sets of measure zero.
4.
That means, in other words, that " T " preserves the concept of a set of measure zero.
5.
So the distributional derivatives are given by the usual derivatives off ? * " ? *, a set of measure zero.
6.
The equality should be understood as holding almost everywhere, that is, differing at most on a set of measure zero.
7.
A set is called a set of full measure if its complement, relative to X, is a set of measure zero.
8.
I'm aware there's some ambiguity for diadic rationals, but a set of measure zero won't matter for this question.
9.
We say that if some property holds at every point of a set of full measure ( or equivalently everywhere except on a set of measure zero ), it holds almost everywhere.
10.
Integration with respect to Lebesgue is independent of sets of measure zero, but integration with respect to the Dirac measure, which is what you are doing, cares about pointwise values.
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