Almost Everywhere

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

$\backslash int\; f(x)\; dx\; \backslash geq\; 0.$

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

for every real numbers a < b, then there exists a null set E (depending on f) such that, if x is not in E, the Lebesgue mean

$\backslash frac\{1\}\{2e\}\; \backslash int\_\{x-e\}^\{x+e\}\; f(t)dt$

converges to f(x) as e decreases to zero. In other words, the Lebesgue mean of f converges to f almost everywhere. The set E is called the Lebesgue set of f.

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

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