Is it standard for a math undergraduate program to have a course on Lebesgue integration?
Yes, and I find it bizarre that a university would not have one.
Lebesgue integration (or measure theory more generally) is necessary for a proper grounding in probability theory. Sure, most statisticians won't need it, but it's a very important part of any pure mathematics course. You certainly can't make a rigorous study of probability or statistics without it.
One answer took "have a course" to mean "is a required course". I would not be surprised if an elementary course on Lebesgue integration would be mandatory (although I would certainly not expect a full measure theory course to be mandatory).
Since you are asking about specific programs: I took a measure theory course in third/fourth year that covered Lebesgue integration (among other things). This is now a second-year course called "Lebesgue Measure and Integration". Additionally, if you take Real Analysis in second year then you cover multivariate Darboux/Riemann integration.
The module guide has this to say about the Lebesgue Measure and Integration course:
Lebesgue Measure and Integration:
Essential for: All Pure Analysis modules; Geometry of curves and
Useful for: All pure mathematics modules. Dynamical Systems;
Statistics modules. Mathematical Finance; Function Spaces and
Applications; Advanced Topics in PDEs; modules involving stochastic
analysis; Random Dynamical Systems and Ergodic Theory.
MATH50006 Lebesgue Measure and Integration
Brief description: Lebesgue's theory of measure and integration is a
powerful extension of the Riemann integral introduced in first-year
analysis. It is an essential tool in all aspects of analysis and its
applications, including probability, stochastic processes and PDEs.
Learning Outcomes: On successful completion of this module, you will be able to:
- explain key features of the construction of Lebesgue measure, and basic features of the measure as a function,
- explain the limitation of a measure, and measurable sets through algebras of open sets,
- define the measurability of functions in terms of measurable sets, and apply simple criteria to verify measurability,
- outline the Lebesgue integral of a function, and how it relates to the Riemann integral,
- work with spaces of functions as infinite dimensional vector spaces, and algebras, and discuss a variety of such spaces,
- use results on convergence of sequences in measurable functions spaces, and criterial to derive properties of limiting functions and
- discuss the relationship between integrability and differentiation,
- explain and use the theorems of Fubini and Tonelli,
- discuss abstract measure theory, and the difficulties with construction of measures and integrals on general spaces.
Module content (syllabus):
An indicative list of sections and topics is:
Motivation: Drawbacks of the Riemann integral, Limits of functions,
Length and area, The Fundamental Theorem of Calculus, Measures of sets
Measure Theory: Abstract measure theory: Motivation, basic definitions
of measure spaces and measures.
Lebesgue Measure in R^d: Volume of rectangles and cubes, The exterior
measure, properties, Lebesgue measurable sets, countable additivity.
Properties of the Lebesgue measure, Regularity, Invariance, σ-algebras
and Borel sets, Non-measurable set.
Measurable functions: Definitions and equivalent formulations of
measurability, Sums and products, compositions, limits of measurable
functions, “Almost everywhere” properties. Approximation by simple
functions and step functions, Egoroff’s and Lusin’s theorems.
Lebesgue integration: Definition using bounded functions on sets of
finite measure, Riemann integrable functions are Lebesgue integrable,
Integrable functions as a normed vector space, L_p(R^d), dense
subsets, Completeness, Fatou’s Lemma, monotone convergence theorem,
uniform integrability, Vitali’s Theorem, Fubini’s and Tonelli’s
Theorems, statements and proofs.
Differentiation and Integration: Differentiation of the Integral,
statement of Lebesgue differentiation theorem, Differentiation of
functions, Functions of bounded variation, properties,
characterisation, Bounded variation implies differentiable a.e.
Absolute continuity of measures, decomposition theorems by Jordan,
Hahn and Lebesgue, Radon-Nikodym Theorem.
I can confirm that when I took this course, all of this was covered (along with some extra material about ergodic theory for the mastery component). This is at Imperial College London.