In Folland's real analysis textbook, he says for two measure spaces $(X, \mathcal{M})$ and $(Y, \mathcal{N})$, and a function $f: X \rightarrow Y$ then $f$ is **measurable** if $f^{-1}(E) \in \mathcal{M}$ for all $E \in \mathcal{N}$. 

He then goes on to discuss properties of integration on $L^+$, the set of all measurable functions $X \rightarrow [0,1]$.

Later, at the start of section 2.3, he starts extending these results to the real valued measurable functions. He says that $f$ (now any real valued function) is **integrable** if $\int |f| < \infty$. He then defines $L^1$ as the space of integrable real-valued functions. 

My question is this, given these definitions, should I be taking all the functions in $L^1$ to be measurable? i.e. given an exercise which asks one to prove some function $f$ is measurable, is it sufficient to show $f$ is integrable? If so, why is measurability necessary in the definition in integrability?