Definition 2

A function f defined on T, with values in $\overline{\mathbf{R}}$ or in a Banach space, is said to be locally $\mu$-integrable if f is $\mu$-measurable and every point $x \in T$ admits a neighborhood V such that $\mu^\bullet(|f|\varphi_V) < +\infty$.

This definition coincides with the one given in Ch. V, §5, No. 1, in case T is locally compact.

Let $f$ be a locally $\mu$-integrable positive function; the mapping $K \mapsto f_K \cdot \mu_K$ is a premeasure (Ch. V, §7, No. 1, Cor. 2 of Th. 1), which will be denoted $f \cdot \mu$.