Example 2

Let us take $X = Y = \mathbf{R}$, and for $\lambda$ and $\mu$ the Lebesgue measure on $\mathbf{R}$ (§1, No. 3); their product is called Lebesgue measure on $\mathbf{R}^2$; the integral of a function $f \in \mathscr{K}(\mathbf{R}^2; \mathbf{C})$ with respect to this measure is denoted $\iint f(x, y)\, dx\, dy$ or $\iint f(x, y)\, dy\, dx$; formula (4), for a function that is zero outside a compact rectangle $[a, b] \times [c, d]$, yields the formula

proved in FRV, II, §3, No. 6.

Since Lebesgue measure on $\mathbf{R}$ is invariant under every translation (§1, No. 3), it follows at once that Lebesgue measure on $\mathbf{R}^2$ is invariant under every translation of $\mathbf{R}^2$.