Proposition 7.4.9: Properties of the Lebesgue Integral
Suppose f and g are two bounded, Lebesgue integrable
functions defined on a measurable set E with finite measure. Then:
-
E c f(x) + d g(x) dx = c
E f(x) dx + d
E g(x) dx
- If A and B are disjoint measurable subsets of
E then
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