Lebesgue Integral - Frank Jones p.201
I'm trying to prove $\Gamma(z)\Gamma(w)=\Gamma(z+w)B(z,w)$ ($z,w$ are complex numbers with positive real part).
If you know an easier way to prove this, please tell me.. :) This property for real variables is easy but i think Fubini is invincible for complex variables.
I will describe a function of which i want to know its measurability.
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Let $z,w\in\mathbb{C}$ such that $Re(z),Re(w)>0$.
Define $F:\mathbb{R}\rightarrow\mathbb{C}$ such that $F(x)=x^{z-1}e^{-x}$ on $(0,\infty)$ and $0$ otherwise.
Define $D=\{(x,y)\in\mathbb{R}^2: 0<x<y\}$
Define $G:\mathbb{R}^2\rightarrow \mathbb{C}$ such that $G(x,y)=(y-x)^{w-1}e^{-(y-x)}$ on $D$ and $0$ otherwise.
Now, define $L(x,y)=F(x)G(x,y), \forall (x,y)\in\mathbb{R}^2$.
How do i prove that $L$ is Lebesgue measurable? Moreover, Lebesgue integrable?