Let
and
are two complete measure spaces. Then there exists a complete measure space
such
that
and

Proposition

(Fubini theorem) Let
and
are two complete measure spaces and the space
is their product as in the previous proposition. Let
be a
-integrable
function. Then

1. Almost surely in
the function
is an
-integrable
function of
.

2. Almost surely in
the function
is an
-integrable
function of
.