Double integration over $y=x$

Question

Let us consider arbitrary, integrable, real function $f$.

The point was my trial to slve the following integral

$$\int_{\mathbb{R}}\int_{\mathbb{R}}f(z,w)dzdw.$$ I expressed is as the sum

$$\int_{\mathbb{R}}\int_{\mathbb{R}}f(z,w)dzdw=\iint\limits_{-\infty<z=w<\infty} f(z,w) dz dw+\iint\limits_{-\infty<z\neq w<\infty} f(z,w) dz dw.$$

To solve the first integral in the previous equation, I employ

$$\iint\limits_{-\infty<z=w<\infty} f(z,w) dz dw= \int_{R} f(z,z) dz.$$

Is this correct?

Answer 1

The straight line has a Lebesgue measure zero. Then the Lebesgue integral over it is zero. The same with Riemann integral - Riemann integrable functions are Lebesgue integrable and both integrals are equal.