Chapter IV

Extension of a measure. $L^p$ spaces

In this chapter, X denotes a locally compact space, $\mu$ a measure on X; when a function is under consideration (absent any specification of the set where the function is defined), it is understood to be a function defined in X.

For every subset A of X, we denote by $\varphi_A$ the characteristic function of A (equal to 1 on A and to 0 on $\mathbf{C}A$).