Additivity of Lebesgue integral

Question

Let $P$ and $C$ measurable sets such that $D = P \cup C$ for some set $D$, suppose $m (P \cap C)= 0$, show the Lebesgue integral over $D$ is equal to the sum of the integrals over $P$ and $C$. When $P \cap C$ is empty, the result is easy defining proper characteristic functions, the problem is when I suppose it is not and my plan is to define a set $A$ of the elements of the intersection so $A$ has measure zero from there I have not found the form to develop the problem.

Answer 1

Let $E=P \cap C$. It follows that $P-E$ and $C-E$ are disjoint. Then, by the result you have proven about the additivity of the integral when the intersection is empty it follows that $\int_{D-E} f d\mu =\int_{P-E} f d\mu +\int_{C-E} f d\mu$.

Now introduce $1_A$, which satisfies $1_A(x)=1$ iff $x \in A$.

Therefore, we have $\int_{X} f1_{D-E} d\mu =\int_{X} f1_{P-E} d\mu +\int_{X} f1_{C-E} d\mu$.

You may now use the Theorem that changing the values of a function on a negligible set does not affect the measurability of a function nor the value of the integral (This might be worth proving) to deduce that: $\int_{X} f1_{D} d\mu =\int_{X} f1_{P} d\mu +\int_{X} f1_{C} d\mu$, which gives $\int_{D} f d\mu =\int_{P} f d\mu +\int_{C} f d\mu$