Almost Everywhere - Properties

Properties

• If f : R → R is a Lebesgue integrable function and f(x) ≥ 0 almost everywhere, then

for all real numbers a < b with equality iff almost everywhere.

• If f : → R is a monotonic function, then f is differentiable almost everywhere.
• If f : R → R is Lebesgue measurable and

for all real numbers a < b, then there exists a set E (depending on f) such that, if x is in E, the Lebesgue mean

converges to f(x) as decreases to zero. The set E is called the Lebesgue set of f. Its complement can be proved to have measure zero. In other words, the Lebesgue mean of f converges to f almost everywhere.

• If f(x,y) is Borel measurable on R2 then for almost every x, the function y→f(x,y) is Borel measurable.

• A bounded function f : -> R is Riemann integrable if and only if it is continuous almost everywhere.