Characteristic function of a measurable set.

Question

Let $X=L^p[0,1]$ $(1\leq p<\infty)$ be the Lebesgue space of p-integrable real functions on $[0,1]$. Let $D\subseteq [0,1]$ be measurable subset. The characteristic functions $\chi_D$ is defined as \begin{equation} \chi_{D}(x)=\begin{cases} 1, & x\in D \\ \\ 0, & x\notin D. \end{cases} \end{equation} How can we define $\chi_Du$, for $u\in M\subset L^p[0,1]$? Where $M$ is a bounded subset of $L^p[0,1]$.

Answer 1

Your $\chi_{D}u$ can be defined in that way: \begin{equation} (\chi_{D}u)(x)=\begin{cases} u(x), & x\in D, \\ \\ 0, & x\notin D. \end{cases} \end{equation}

Edit: You don't need to mention that $u\in M\subseteq L^p[0,1]$ where $M$ is a bounded subset of $L^p[0,1]$. This is equivalent to $u\in L^p[0,1]$ without mentioning $M$ at all because for each $u\in L^p[0,1]$ you can choose $M$ to be $M:=\{u\}$.