Question

I am teaching myself measure theory from Bartles "The elements of integration and lebesgue measure". In order to prove Riesz Representation Theorem he defines a set function $\lambda(E) = G(1_E)$. Where $G$ is a linear bounded functional, and $1_E$ is the usual characteristic function. In order to show that $\lambda$ is absolutely continuous with $\mu$, He claims that $\lambda$ defined as before is zero on a set of measure zero. We are working under a measure space $(X, \sigma(X), \mu)$ and $G$ is a bounded linear functional on $L_1(X, \sigma(X), \mu)$. The question is why is $\lambda(M) = G(1_M)$ zero if $M$ is a set of measure zero? I am unable to figure out how this comes about from the fact I know about linear functionals $G(af + bg) = aG(f) + bG(g)$ where $f,g \in L_1$.