Sum of Integrals on Complementary Sets

Theorem
Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

Let $A, E \in \Sigma$ with $A \subseteq E$.

Let $f$ be a $\mu$-integrable function on $X$.

Then
 * $\displaystyle \int_E f \ \mathrm d \mu = \int_A f \ \mathrm d \mu + \int_{E \mathop \setminus A} f \ \mathrm d \mu$

Proof
Let $\chi_E$ be the characteristic funtion of $E$.

Because $A$ and $E \setminus A$ are disjoint:
 * $A \cap \left({E \setminus A}\right) = \varnothing$

By Characteristic Function of Union:
 * $\displaystyle \chi_E = \chi_A + \chi_{E \mathop \setminus A}$

Then by Multiplication of Numbers Distributes over Addition:
 * $f \chi_E = f \chi_A + f \chi_{E \mathop \setminus A}$

Integration over $E$ gives:

Also see

 * Sum of Integrals on Adjacent Intervals