Summary: 1. The Theorems of Fubini and Tonelli.
Suppose n, m N+
and 0 < m < n. We identify Rn
with Rm
× Rn-m
in the
natural way.
For each function f with domain Rn
and each x Rm
we let
sx(f) = {(y, f(x, y)) : y Rn-m
};
thus sx(f) is a function with domain Rn-m
such that
sx(f)(y) = f(x, y) whenever y Rn-m
.
Lemma 1.1. Suppose f F+
n and
F(x) = ln-m(sx(f) for x Rm
.

Source: Allard, William K. - Department of Mathematics, Duke University

Collections: Mathematics