The convolution of two locally Lebesgue integrable functions is defined by (1.2). for for which the integral exists.
Accordingly, for any random variable X, E X) is defined to be the Lebesgue integral of X with respect to the probability measure P, provided that the integral exists.
provided the integral exists.
for all for which the integral exists.
whenever the double integral exists and is finite.
then the Riemann-Stieltjes integral exists and the following inequality holds: (2.13).
provided that the integral exists on, where is the Gamma function.
for u ∈ D{f}, where D{f} consists of all complex numbers u ∈ R for which the improper integral exists.
where Γ is the standard gamma function defined by Γ = ∫ 0 ∞ e − s s α − 1 d s, provided the integral exists, where log = log e.
for z ∈ Ω{f}, where Ω{f} consists of all complex numbers z ∈ R for which the improper integral exists, (see [11 15]).
It is well known that if is a bounded function, is of bounded variation, and the Riemann-Stieltjes integral exists, then the following inequality holds: (2.7).
