Measure Theory

Measure theory is a branch of mathematical analysis that assigns generalized sizes to sets and develops integration with respect to those sizes. A measure can represent length, area, volume, the number of elements, or probability. By specifying which sets are measurable and requiring countable additivity, the theory provides a common framework for discrete and continuous settings. It underlies modern probability and extends the integration methods of calculus to broader classes of functions and spaces. (terrytao.files.wordpress.com)

Historical development

Modern measure theory emerged from nineteenth-century work on integration and the structure of sets. Émile Borel introduced a countably additive approach to measure in lectures published in 1898. Henri Lebesgue subsequently connected the measurement of sets with a new theory of integration, presented in his 1902 doctoral dissertation, Intégrale, longueur, aire. These developments addressed limitations of Riemann integration, particularly in handling discontinuous functions and limiting operations. (mathshistory.st-andrews.ac.uk)

Measure spaces and countable additivity

The basic structure is a measure space $(X,\Sigma,\mu)$. Here $X$ is a set, $\Sigma$ is a sigma-algebra of subsets of $X$, and $\mu$ assigns each member of $\Sigma$ a value in $[0,\infty]$. The pair $(X,\Sigma)$ is called a measurable space. A sigma-algebra contains $X$ and is closed under complements and countable unions; its members are the measurable sets. (math.mit.edu)

A measure satisfies $\mu(\varnothing)=0$ and, for every sequence of pairwise disjoint measurable sets $A_1,A_2,\ldots$,

This requirement is stronger than finite additivity. It implies monotonicity and countable subadditivity. For increasing measurable sets $A_n$, the measures converge to the measure of their union. For decreasing sets, convergence to the measure of their intersection holds when the first set has finite measure. These properties make measures compatible with countable limiting operations. (math.mit.edu)

Examples and measurable sets

Lebesgue measure on the real line assigns an interval its usual length; in higher-dimensional Euclidean spaces it generalizes area and volume. Counting measure assigns a finite set its number of elements and an infinite set infinite measure. A Dirac measure concentrated at a point assigns a set measure one if it contains that point and zero otherwise. These examples illustrate that measurement need not depend on geometric extent. (terrytao.files.wordpress.com)

On a space equipped with a topology, the Borel sets form the sigma-algebra generated by open sets. Lebesgue measurable sets include the Borel sets and every subset of a Borel set of Lebesgue measure zero. A measure is complete when every subset of a measurable null set is itself measurable. (math.mit.edu)

Not every subset of Euclidean space can receive a measure consistent with ordinary volume, translation invariance, and countable additivity. The Vitali construction demonstrates this obstruction on the real line. Consequently, distinguishing measurable from nonmeasurable sets is an essential part of the theory, rather than merely a technical convention. (math.mit.edu)

Measurable functions and integration

A measurable function respects the measurable structures of its domain and target: inverse images of measurable target sets must be measurable. For real-valued functions, it suffices to test inverse images of Borel sets. This condition ensures that sets defined by the function’s values can be assigned measures. (math.mit.edu)

The Lebesgue integral begins with nonnegative simple functions, which take finitely many values on measurable sets. If

where the $A_k$ are disjoint and $a_k\geq0$, then

with $0\cdot\infty=0$. For a nonnegative measurable function $f$, its integral is the supremum of the integrals of simple functions satisfying $0\leq s\leq f$. General real-valued functions are treated through positive and negative parts; integrability requires $\int|f|\,d\mu<\infty$. (math.mit.edu)

A property holds almost everywhere if its exceptions lie in a measurable set of measure zero. Integrable measurable functions that agree almost everywhere have equal integrals, allowing many analytical statements to disregard null-set differences. (math.mit.edu)

Convergence and product measures

Several central theorems govern interchange of integration and limits. The monotone convergence theorem applies to increasing sequences of nonnegative measurable functions. Fatou’s lemma bounds the integral of a lower limit by the lower limit of the integrals. The dominated convergence theorem permits passage of a limit through an integral when almost-everywhere convergence is controlled by a single integrable dominating function. Their hypotheses are essential: pointwise convergence alone does not guarantee convergence of integrals. (math.mit.edu)

Product measures extend measurement to Cartesian products. For sigma-finite measure spaces, Tonelli’s theorem permits iterated integration of nonnegative measurable functions, even when the integral is infinite. Fubini’s theorem permits iterated integration and interchange of order for integrable functions. These results connect multidimensional integration with successive one-dimensional integrations. (math.mit.edu)

Densities, probability, and function spaces

A measure $\nu$ is absolutely continuous with respect to $\mu$ when every $\mu$-null set is also $\nu$-null. For sigma-finite positive measures, the Radon–Nikodym theorem gives a measurable density, the Radon–Nikodym derivative, satisfying

The density is unique up to equality almost everywhere. A probability density function is a particular instance of this construction. (math.mit.edu)

A probability space is a measure space whose total measure is one. A random variable is a measurable function, its distribution is the induced measure on its target space, and its expectation is an integral. This language treats discrete, continuous, and mixed distributions within one framework. (math.mit.edu)

Measure theory also defines $L^p$ spaces by grouping measurable functions that agree almost everywhere and controlling the integrals of their powers. For $1\leq p<\infty$, the norm is $(\int|f|^p\,d\mu)^{1/p}$. These are complete normed spaces; $L^2$, with its integral inner product, is a Hilbert space. They provide fundamental settings for Fourier transforms and analytical estimates. (math.mit.edu)