Example 1

Let X be a locally compact topological space and let $\mu$ be a positive measure on X. Let $p_1$ and $p_2$ be fixed elements of $[1,+\infty [$.

Let $g$ be a $\mu$-measurable function on X with values in K. Let D denote the subspace of $\mathscr{L}_K^{p_1}(X, \mu)$ formed by the functions $f$ in $\mathscr{L}_K^{p_1}(X, \mu)$ such that $gf\in \mathscr{L}_K^{p_2}(X, \mu)$. The linear mapping from D into $\mathscr{L}_K^{p_2}(X, \mu)$ defined by $f\mapsto gf$ determines a partial operator from $\mathscr{L}_K^{p_1}(X, \mu)$ into $\mathscr{L}_K^{p_2}(X, \mu)$, which is denoted by $m_g$.

The vector subspace of $\mu$-negligible functions in $\mathscr{L}_K^{p_1}(X, \mu)$ is contained in D, and the image under $m_g$ of a $\mu$-negligible function is again $\mu$-negligible. We shall denote by $\widetilde{m}_g$ the partial operator from $L^{p_1}_K(X, \mu)$ into $L^{p_2}_K(X, \mu)$ deduced from $m_g$ by passing to quotients. It is said to be the multiplication operator by $g$ from $L^{p_1}_K(X, \mu)$ into $L^{p_2}_K(X, \mu)$. Functions $g_1$ and $g_2$ which are locally equal $\mu$-almost everywhere define the same multiplication operator.