I am having difficulty in evaluating the following integral: $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x) f(y) f(z)\chi_{{(x,x,x): x\in \mathbb{R}}} dx dy dz$.
Could someone please give me some hint or tell me how to approach?
Is it $(\int_{-\infty}^{\infty} f(x) dx)^3$ or $\int_{-\infty}^{\infty} f(x)^3 dx?$
Any help or hint would be appreciated. Thanks in advance.
Assuming that you may interchange integrals it is $$\int f(x)^3 dx,$$ since the indicator function integrated over $y$ and $z$ is evaluating the function so that $$\int\int f(y)f(z)\mathbb{1}_{\{x,x,x\}}dydz=f(x)f(x).$$