Definition 1

The functions belonging to $\mathscr{L}_F^1(X, \mu)$ are said to be integrable with respect to the measure $\mu$ (or, again, that they are $\mu$-integrable). *The integral (with respect to $\mu$) of the integrable function $f$ is by definition the value at $f$ of the extension by continuity to $\mathscr{L}_F^1$ of the linear mapping $g \mapsto \int g\, d\mu$ of $\mathscr{H}_F$ into $F$; it is again denoted $\mu(f)$ or $\int f\, d\mu$, or $\int f(x)\, d\mu(x)$ or $\int f\mu$, or $\int f(x)\mu(x)$.