## 7.4. Lebesgue Integral

### Proposition 7.4.10: Properties of the Lebesgue Integral

Suppose

*f*and*g*are two bounded, Lebesgue integrable functions defined on a measurable set*with finite measure. Then:***E**-
_{E}c f(x) + d g(x) dx = c_{E}f(x) dx + d_{E}g(x) dx - If
and**A**are disjoint measurable subsets of**B**then**E**

_{A B}f(x) dx =_{A}f(x) dx +_{B}f(x) dx - If
*f(x) = g(x)*for all*x*in**E***except possibly on a set of measure zero*then_{E}f(x) dx =_{E}g(x) dx - If
*f(x) g(x)*for all*x*in**E***except possibly on a set of measure zero*then_{E}f(x) dx_{E}g(x) dx