We have from an application of the Fubini theorem
that (47) generalizes the classical Fourier transform.
For x [member of] [L.sub.p]([R.sup.+], X), by Fubini Theorem
and condition (b) of Theorem 2.1 of , we obtain
Setting u = [e.sup.ax+by] in (10), in view of Fubini theorem (cf.
For [[sigma].sub.1] = [sigma], by Fubini theorem (see ) and (29), we have
and then by the use of the stochastic Fubini Theorem
again we have
Making use of the Fubini theorem
and a substitution technique, we have
If, in addition, dimX < [infinity], then a classical Fubini theorem
allows one to conclude that Z is of Lebesgue measure zero.
The proof is a simple application of the Fubini theorem
In view of the Fubini theorem
, we have [J.sub.1] = [K.sub.1].
The techniques are a melange of Fubini theorem
, an elementary version of resolution of singularities and some fairly standard results from the theory of generalized functions.
by the Fubini theorem
and the change of variable theorem, which proves (60).
According to the law of iterated expectations and Fubini theorem