Theorem 2

Let $\lambda, \mu$ be two measures defined, respectively, on two locally compact spaces $X, Y$. For every function $f$ in $\mathscr{K}(X \times Y; \mathbf{C})$, the integral of $f$ with respect to the product measure $\nu$ of $\lambda$ by $\mu$ has the value

Because of the last formula, the integral of $f$ with respect to the product measure $\nu$ is usually denoted $\iint f\, d\lambda\, d\mu$, or $\iint f\, d\mu\, d\lambda$, or $\iint f\, \lambda\mu$, or $\iint f\, \mu\lambda$, or $\iint f(x, y)\, d\lambda(x)\, d\mu(y)$, or $\iint f(x, y)\, d\mu(y)\, d\lambda(x)$, or $\iint f(x, y)\, \lambda(x)\mu(y)$, or $\iint f(x, y)\, \mu(y)\lambda(x)$; it is said to be the double integral of $f$ with respect to $\lambda$ and $\mu$. With this notation, the formula (3) may be written

Formula (3) shows that if $\lambda$ and $\mu$ are real (resp. positive) measures, then the product measure $\nu$ is real (resp. positive).