I could see people working with rigorous measure theory in QM and mathematical QFT using Lebesgue integrals as a convenient mathematical formalism, but it is not as useful of a computational tool like the Riemann integral is.
Properties. Given a set E of real numbers, µ(E) will denote its Lebesgue measure if it’s defined. I have been working through measure theory, specifically the dominated convergence of Lebesgue integrals and its applications such as differentiating under the integral sign. There I came across the .

It should be noted that it assumes familiarity with sequences, series, limits, continuity, and compactness at the level of a first undergraduate course in real analysis, so it is not a self-contained text. This module aims to introduce Lebesgue's theory of measure and integration, which extends the familiar notions of volume and "area under a graph" associated with the Riemann integral. In summary, Lebesgue measure is the extension of volume from boxes to a large class of subsets of Rd on which requirements (a)–(d) are satisfied.

However, while Lebesgue measure is the prototypical example of a measure on Rd, it is not the only way to measure things. It is sufficiently to recall that in the probability theory, a Borel measurable application is also a random variable defined on a selection space.

Rectangles, Whitney decomposition, Exterior Lebesgue measure and its properties (monoton, countable subadditive), measurable sets (Lebesgue measurable sets are a σ-algebra, compare the summary and characterisation of measurability), F σ,G δ-sets 1.2. Measure theory is now applied to most areas of mathematics such as functional analysis, geometry, probability, dynamical systems and other domains … So to answer your question, I don't know of any application outside of being a formal tool. 1. The Lebesgue measure (L) is of great importance in applications on Rn Lebesgue measure is frequent used in problems of the probability theory, in physics and other domains. A User-Friendly Introduction to Lebesgue Measure and Integration is based on course notes written specifically for, and as the author explains in the preface, by undergraduate students. This is Lebesgue measure on Rd. Stack Exchange Network. where measure theory evolved via the spectral theory of operators on Hilbert space to von Neumann algebras with applications to diverse parts of mathematics & physics. Here are the properties we wish it to have. Although the Lebesgue measure was initially developed in Euclidean spaces, the theory is independent of the geometry of the background space and applies to abstract spaces as well.