Change of Variable, Weightings

This chapter shows that the Lebesgue spaces, L p (Ω;E) are preserved under a regular change of variables. It explains that the regrouping of variables, and hence also their separation, preserves the space L p (Ω;E) when p is finite, but that this is not the case when p is infinite. The chapter explains permutation of the variables of a p -integrable measure of q -integrable measures. It defines the weighting of two measures. The chapter verifies that users recover the weighting by a compactly supported weight, which justifies the use of the same notation.