existence of a double integral $ f(x,y)=\left\{ \begin{array}{c} 1&;&x=y\\ 0&;&x\neq y \end{array} \right. $

Question

Let $f$ a function defined in $Q=[0,1]\times[0,1]$,

$ f(x,y)=\left\{ \begin{array}{c} 1&;&x=y\\ 0&;&x\neq y \end{array} \right. $

I have tried to demonstrate the existence of $\displaystyle\iint_Q f$ and also that it is equal to zero. I tried to see that the set of discontinuity points of the function has a null measure, but it doesn't work. Are there any idea?

Answer 1

Observe that $f$ is bounded. Moreover $f$ is Riemann integrable, since the discontinuity point set is Lebesgue measurable and has zero Lebesgue measure.

The calculation of the integral is simple, you could try to solve it.

Answer 2

To prove that the diagonal set $S = \{(x,y), x=y\}$ is of measure $0$, you can say that for each $n$, it is included in the union of squares $C_{k,n}$ of center $(k/n,k/n)$ and side of size $1/n$. Thus, the Lebesgue measure of this set, $|S|$, verifies $$ |S| \leq \sum_{k=1}^{n} |C_{k,n}| = \sum_{k=1}^{n} \frac{1}{n^2} = \frac{1}{n} $$ which converges to $0$ when $n\to\infty$. Therefore $S$ is of measure $0$. And you easily get that the integral is $0$.