almost everywhere in the interval , then there exist and satisfying such that almost everywhere in and almost everywhere in
As a corollary, if the integral above is 0 for all then either or is almost everywhere 0 in the interval Thus the convolution of two functions on cannot be identically zero unless at least one of the two functions is identically zero.
As another corollary, if for all and one of the function or is almost everywhere not null in this interval, then the other function must be null almost everywhere in .
The theorem can be restated in the following form:
Let . Then if the left-hand side is finite. Similarly, if the right-hand side is finite.
Above, denotes the support of a function f (i.e., the closure of the complement of f-1(0)) and and denote the infimum and supremum. This theorem essentially states that the well-known inclusion is sharp at the boundary.
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