Measure and integration

A sigma algebra specifies which sets may be measured; it contains the whole space and is closed under complements and countable unions. A measure assigns these sets nonnegative sizes and adds sizes across countably many disjoint pieces. Lebesgue measure agrees with ordinary interval length. A probability measure instead assigns total mass one. These are the same structural rules with different interpretations, so an expectation is an integral over a probability space rather than a separate mathematical species.

A null set has measure zero. Each singleton in the real line has zero Lebesgue measure, so countable additivity makes the rationals null, even though every open interval contains rational points. Density concerns proximity; measure concerns size. Saying a property holds almost everywhere means its exceptional set is null for the specified measure. A point mass probability distribution can give one point probability one, so a Lebesgue-null set is not automatically null for every other measure.

For a nonnegative simple function, multiply each value by the measure of its level set and add. The nonnegative Lebesgue integral is obtained by approximating from below with such simple functions. A signed or complex function is integrable when its absolute value has finite integral. This absolute-integrability requirement prevents ambiguous subtraction of two infinities. It also explains why some oscillatory physics integrals require a regulator or distributional interpretation instead of being ordinary Lebesgue integrals.

The dominated convergence theorem provides a powerful limit rule. Suppose measurable functions converge pointwise almost everywhere and a single integrable function g bounds their absolute values almost everywhere. Their integrals then converge to the integral of the limit. The word single is essential: a bound that changes with n and grows without control proves nothing. Monotone convergence is a different theorem for increasing nonnegative sequences and does not require an integrable dominating function.

For finite p≥1, the Lp norm measures an integrated power of a function's magnitude. Functions that differ only on a null set have zero Lp distance, so elements of Lp are equivalence classes rather than functions with uniquely specified point values. This is useful for energy and mean-square error, but it means evaluation at one point is not generally a well-defined operation on an L2 equivalence class. Continuity or a stronger function space may restore meaningful point evaluation.

Convergence almost everywhere does not ensure convergence of integrals. A spike can grow taller while narrowing fast enough to disappear at each fixed positive point yet retain fixed area. This is the mechanism behind many dangerous exchanges of limits. The same warning appears in machine learning when rare events carry large losses: looking typical point by point does not control expected loss. An integrability or uniform-integrability argument is needed to rule out escaping mass.