Lebesgue integral

The Lebesgue integral assigns values to measurable functions by measuring the sizes of their level sets. It extends integration beyond functions handled by elementary Riemann integration and applies on general measure spaces.

Measure space: A set equipped with a collection of measurable subsets and a measure assigning them sizes. The integral is defined relative to this underlying structure.
Riemann integral: An integral defined through limits of sums over partitions of a domain. Its partition-based approach contrasts with measuring function level sets.
Lebesgue spaces: Function spaces whose members have finite integrals of specified powers of their absolute values. Their norms are built directly from Lebesgue integrals.
Darboux integral: An integral defined by matching upper and lower sums over partitions of a domain. It characterizes Riemann integrability through bounds on intervals, unlike level-set integration.
Measurable function: A function whose inverse images of measurable sets are measurable. Measurability ensures the function's level sets can be measured.
Almost everywhere: A property holds almost everywhere when the set where it fails has measure zero. Changing a function on a null set does not change its Lebesgue integral.
Expected value: The probability-weighted average of a random variable, defined by integration against its probability distribution. Expected values are Lebesgue integrals on probability spaces.
Henstock–Kurzweil integral: An integral defined by gauge-controlled tagged sums that integrates some functions not Lebesgue integrable. It extends integrability to certain conditionally convergent functions beyond the Lebesgue class.
Simple function: A measurable function taking only finitely many values. Lebesgue integration begins by integrating these finite-valued approximations.
Null set: A measurable set with measure zero. Null sets explain why pointwise differences can be irrelevant to integration.

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