Hi I am confused on something. We defined the lebesgue integral of a general f in the following way: If $\int f^+ <\infty $ and $\int f^- < \infty$ then $\int f= \int f^+ - \int f^-. $
So basically if we have an integral like $\int f+g $ then we can only seperate it if both f and g are integrable?
Also if we have a certain integral of f over the real numbers, can we split that integral into the sum of integrals over sets $A_i$ whose union is the real numbers? Can we do this even if we dont know whether the function is integrable? Thanks!
Note that functions that aren't lebesgue integrable may still sum to a lebesgue integrable function: just take any non-integrable $f$ and let $g = -f$; neither are lebesgue integrable, but their sum is the zero function, which is lebesgue integrable.
Furthermore, if $f$ is a complex valued lebesgue integrable map on a measure space $(X,\mathbf{M},\mu)$, then the map
$\tilde{\mu}: \mathbf{M} \to \mathbb{C} \,;\, E \mapsto \int_E fd\mu$
will be a complex measure on $\mathbf{M}$, which is to say that it is complex valued and countably additive. Consequently, if $A_1,A_2,... \in \mathbf{M}$ are pairwise disjoint measurable sets then
$\int_{\bigcup_iA_i} f d\mu = \tilde{\mu}\left( \bigcup_i A_i \right) = \sum_i \tilde{\mu}(A_i) = \sum_i \int_{A_i} f d\mu$