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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:
  1. E c f(x) + d g(x) dx = c E f(x) dx + d E g(x) dx
  2. If A and B are disjoint measurable subsets of E then
    A B f(x) dx = A f(x) dx + B f(x) dx
  3. 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
  4. 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
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