With the Lebesgue integral, it becomes easy to consider the Gamma function and the theory of Laplace
transforms. I will use the standard notation for the integral used in calculus, but remember that all
integrals will be Lebesgue integrals taken with respect to one dimensional Lebesgue measure. First
is a very important function defined in terms of an integral. Problem 14 on Page 552 shows
that in the case of a continuous function, the Riemann integral and the Lebesgue integral are
exactly the same. Thus all the standard calculus manipulations are valid for the Lebesgue
integral provided the functions integrated are continuous. This also implies immediately that
the two integrals coincide whenever the function is piecewise continuous on a finite interval.
Recall that the value of the Riemann integral does not depend on the value of the function at
single points and the same is true of the Lebesgue integral because single points have zero
measure.

Definition 10.1.1The gamma functionis defined by

∫
∞ −t α−1
Γ (α) ≡ 0 e t dt

whenever α > 0.

Lemma 10.1.2The integral is finite for each α > 0.

Proof: By the monotone convergence theorem, for n ∈ ℕ