A combination of (50), (53) and

Fubini's theorem yields the relation

Applying

Fubini's theorem for the first part of (2.

Then, by the Lemma, Lebesgue dominated convergence theorem, and Fubini's theorem,

Let u = x - t + y/2 and v = x - y/2 in the last term, by Lemma, Fubini's Theorem and the Lebesgue dominated convergence theorem,

Hence, using

Fubini's theorem and the first point of Proposition 10,

For the first expectation, by

Fubini's theorem we have

1) and

Fubini's theorem we have for almost every y [member of] I[R.

Then, the

Fubini's theorem implies that we can exchange the order of the sum and the q-integral signs, and we obtain

Using

Fubini's theorem we can rewrite the right-hand side of Eq.

Since [0, +[infinity][x{0} and {0} x [0,+[infinity][ are [mu]-nulls, by

Fubini's theorem and a change of variable we obtain

Hence, using the

Fubini's theorem and the properties of the generalized q-Dunkl translation, we get

So, by the

Fubini's theorem, we can exchange the order of the q-integrals and obtain,