Sum of Integrals on Complementary Sets

Theorem
Let $\struct {X, \Sigma, \mu}$ be a measure space.

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

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

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

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

Because $A$ and $E \setminus A$ are disjoint:
 * $A \cap \paren {E \setminus A} = \O$

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

Integrating $f$ over $E$ gives:

Also see

 * Sum of Integrals on Adjacent Intervals for Integrable Functions