Let $(X, \mathcal S, \mu)$ be a measure space. Let $\Bbb L$ be the collection of all $\mathcal S$-measurable functions and let $L_1(\mu)$ be the collection of all $\mu$-integrable functions i.e. the collection of all the functions $f : X \longrightarrow \Bbb R^*$ such that $\int f^+\ d\mu < +\infty$ and $\int f^-\ d\mu < +\infty,$ where $f^+$ and $f^-$ respectively denote the positive part and the negative part of the function $f.$ Now suppose that $f \in L_1(\mu).$ Can we say that $f \in \Bbb L?$ What I know is that if $f \in \Bbb L$ and $\int f^+\ d\mu, \int f^-\ d\mu < +\infty$ then $f \in L_1(\mu).$ Is the converse true?
Any help in this regard will be highly appreciated. Thanks in advance.
Yes. Remember how the integral was defines. First for simple and of course measurable functions. measurability is necessary here or course. Then, for positive (measurable) functions, you basically construct a sequence of simple measurable functions using preimages of the positive functions. This is where -maybe silently- the measurability condition is used.