The Riemann-Stieltjes integral and measure extend the Riemann integral to functions of bounded variation. A Riemann-Stieltjes measure is a signed measure that corresponds to the integral. Stieltjes functions are functions with bounded variation that generate Riemann-Stieltjes measures. These concepts play a vital role in probability theory, vector calculus, and stochastic processes. Embark on a Riemann-Stieltjes … Read more
The Borel sigma algebra, denoted as “B(X)”, is a collection of sigma-algebras that is defined on a set X and includes all subsets of X that can be formed by countable unions, intersections, and complements of Borel sets. Borel sets are sets that are obtained from open sets through a series of countable set operations. … Read more
The Borel sigma field, denoted by σ(B), is a collection of sets that contains all open intervals, half-open intervals, closed intervals, and their complements. It is the smallest sigma field that contains all open sets of the real line. The Borel sigma field is important in measure theory because it allows for the definition of … Read more
Almost sure convergence, denoted as a.s. convergence, is a property of a sequence of random variables that converges to a limit with probability 1. This means that the probability of the event that the sequence converges to the limit is equal to 1, or in other words, the convergence happens “almost surely” or “with probability … Read more
Convergence in measure is a type of weak convergence that describes the behavior of a sequence of random variables as their indices increase. It quantifies how close the probability distributions of the random variables are to each other, in terms of their cumulative distribution functions. Specifically, convergence in measure means that for any fixed set … Read more
Inner measure is a concept in measure theory that quantifies the size of a set in a way that is closely related to the Lebesgue measure. It is defined as the supremum of the measures of all closed sets contained in the set. This definition highlights the relationship between the inner measure and the Lebesgue … Read more
Algebra and Sigma Algebra In measure theory, sigma algebras play a crucial role in defining measurable sets and defining probability spaces. They generalize the concept of an event space in probability theory and provide a framework for constructing measures that assign probabilities or other values to sets of outcomes. Sigma algebras are built from sets … Read more
The Lebesgue Dominated Convergence Theorem is a fundamental convergence theorem in Lebesgue integration theory. It states that if a sequence of measurable functions f_n converges pointwise almost everywhere to a function f, and if there exists an integrable function g such that |f_n| ≤ g for all n, then the sequence of integrals ∫f_n dx … Read more
The Borel covering lemma, attributed to Émile Borel, provides a fundamental result in measure theory. It states that given a regular Borel measure and an open set, there exists a countable collection of Borel sets that covers the open set with arbitrarily small total measure. This lemma plays a crucial role in various measure-theoretic constructions, … Read more
A Lebesgue measurable function is a function whose domain and range are both Lebesgue measurable sets. A Borel measurable function is a function whose domain and range are both Borel sets. Lebesgue measurable functions are more general than Borel measurable functions, as all Borel sets are also Lebesgue measurable. Both types of functions are important … Read more
Measure equivalence describes the relationship between two measures that share the same sets of negligible size. In measure theory, it categorizes measures as absolutely continuous, mutually singular, or equivalent. These measures find applications in probability theory, analysis, and ergodic theory. Understanding measure equivalence allows for the comparison and transformation of measures, providing insights into probabilistic … Read more
The Borel sigma algebra is a collection of sets that defines the measurable events in a probability space. It is generated by the open sets of the sample space and is the smallest sigma algebra that contains all open sets. The Borel sigma algebra is named after Émile Borel, a French mathematician who made significant … Read more
