Together with (3.5) in [1],

Fubini's theorem, and the reproducing property, we can see that

By law of total expectation and

Fubini's theorem, we have

Combining (2.18)-(2.19) and

Fubini's theorem, when [x.sub.1] [conjunction] [x.sub.2] [greater than or equal to] [x'.sub.3] and t > 0, we have

Making the change of variables y = x + [tau]/2 and z = x - [tau]/2 and applying

Fubini's theorem we obtain

Using the assumption that the QFT of quaternion mother wavelet is real-valued and then applying

Fubini's theorem, we obtain

Applying change of variables and

Fubini's theorem, we can derive the following four estimates:

A combination of (50), (53) and

Fubini's theorem yields the relation

Applying

Fubini's theorem for the first part of (2.19), we get

Then, by the Lemma, Lebesgue dominated convergence theorem, and

Fubini's theorem,

Hence, using

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

since E [[integral].sup.t.sub.0] B[(t).sup.4] dt = [[integral].sup.T.sub.0] E[B[(t).sup.4]] dt by

Fubini's theorem and E[B[(t).sup.4]] = 3[t.sup.2].

From the relation (4.1) and

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