Measure Theory is the branch of mathematical analysis that studies measures, which assign a consistent notion of size, length, area, volume or probability to subsets of a space. It provides the rigorous foundation for the Lebesgue integral, which generalises the Riemann integral and handles a wider class of functions and limiting operations. It is the formal basis of modern probability theory, where a probability is a measure of total mass one.
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- A measure is a function defined on a sigma-algebra of subsets that is non-negative, assigns zero to the empty set and is countably additive, meaning the measure of a countable disjoint union equals the sum of the measures. This structure makes precise which sets are measurable and how their sizes combine, resolving paradoxes that arise when trying to measure all subsets of a space.
- Building on measures, the Lebesgue integral defines integration by partitioning the range of a function rather than its domain, which gives strong convergence theorems such as the monotone and dominated convergence theorems. These results justify exchanging limits and integrals under mild conditions, a frequent need in analysis.
- Measure theory underpins modern probability, where outcomes form a measurable space, events are measurable sets and random variables are measurable functions. This framework supports the rigorous treatment of expectation, conditional probability and the stochastic processes used across the sciences and quantitative finance.