The intersection of Lebesgue measurable sets is Lebesgue measurable and a countable union of Lebesgue measurable sets is Lebesgue measurable since the collection of Lebesgue measurable sets is a $\sigma$-algebra.

Integration" , Addison-Wesley (1975) pp. If (,) and (,) are two measurable spaces, then a function : → is called measurable if for every Y-measurable set ∈, the inverse image is X-measurable – i.e. [Bor] E. Borel, "Leçons sur la theorie des fonctions" , Gauthier-Villars (1898) Zbl 29.0336.01 [Bou] N. Bourbaki, "Elements of mathematics. Chapt.6;7;8 (Translated from French) MR0583191 Zbl 1116.28002 Zbl 1106.46005 Zbl 1106.46006 Zbl 1182.28002 Zbl 1182.28001 Zbl 1095.28002 Zbl 1095.28001 Zbl 0156.06001 [Hal] In this setup, the composition of measurable functions is measurable, making the measurable spaces and measurable functions a category , with the measurable spaces as objects and the set of measurable functions as arrows. BEST APPROXIMATION PROPERTIES IN SPACES OF MEASURABLE FUNCTIONS 3 Define by (T,Σ,µ) a measure space and by L0(T) the set of all (equivalence classes of) extended real valued µ measurable functions on T. For simplicity we use the short notation L0 = L0([0,α)) with the Lebesgue measure m on [0,α), where α = 1 or α = ∞. Measure Theory and Fine Properties of Functions, Revised Edition provides a detailed examination of the central assertions of measure theory in n-dimensional Euclidean space. The book emphasizes the roles of Hausdorff measure and capacity in characterizing the fine properties of sets and functions. Measurable Functions and their Integrals 1 General measures: Section 10 in BillingsleyÞ Recall: a probability measure on a -field on a space is a real-T 5Y H valued function on with the properties:Y (a) TÐÑœ!9 (b) TÐÑœ"H (c) if are disjoint.TÐ EÑœ TÐEÑ E 3œ" _ 333 3œ" _! : (−) ∈.