Convolution and representations

Abstract

Recall (Ch. V, §6, Nos. 1 and 4; Ch. VI, §2, No. 10) that, if X and Y are locally compact spaces, μ a measure on X, and φ a mapping of X into Y, φ is said to be μ-proper if: a) φ is μ-measurable; b) for every compact subset K of Y, ^φ₋₁ (K) is essentially μ-integrable. Then the image measure v = φ(μ) on Y exists and has the following property: for a function f on Y, with values in a Banach space or in $\overline R$, to be essentially integrable for v, it is necessary and sufficient that f ○ φ be so for μ, in which case,

$$ \int_Y {f\left( y \right)} dv\left( y \right) = \int_X {f\left( {\varphi \left( x \right)} \right)} d\mu \left( x \right) $$