Indicator Functions on Measureable Sets are in L+
# Statement
Let $(X, \mathcal{M})$ be a Measure Space. Suppose $A \in \mathcal{M}$ . Then $1_{A} \in L^{+}(\mathcal{M})$.
# Proof
Indicator Functions have by definition the Codomain ${0, 1}$, so they are non-negative. Thus, it suffices to check that $1_{A}$ is $\mathcal{M}$-measureable. Checking the definition of Measureable Function and recalling the form of a Sigma Algebra induced by an Indicator Function: $$\sigma(1_{A}) = {\emptyset, A, A^{C}, X} \subset \mathcal{M}$$
since $A \in \mathcal{M}$. Thus $1_{A}$ is $\mathcal{M}$-measureable. $\blacksquare$