needed to answer it, from measure theory, integration theory, some functional analysis, to some representation theory. Suppose that every bounded subset of E has outer measure zero. Real Analysis 1, MATH 5210, Fall 2018 Homework 5, Lebesgue Outer Measure and The σ-Algebra of Lebesgue Measurable Sets, Solutions 2.7. Additivity of separated sets 5. 1 MeasureTheory The sets whose measure we can define by virtue of the preceding ideas we will call measurable sets; we do this without intending to imply that it is not possible 7 • 3. m∗ is countably subadditive (Proposition 2.3). Prove that for any bounded set E, there is a Gδ set G for which E ⊂ G and m∗(G) = m∗(E). Monotonicity 2. Show that if a set Ehas positive outer measure, then there is a bounded subset of Ethat also has positive outer measure. In conclusion, outer measure m∗ satisfies: 1. 4 Properties of outer Jordan content In this section we compare the properties of the exterior measure m (E) with the outer Jordan content J (E). (21) We have the following theorem that states that the outer measure fulfills several desirable properties for the notion of a size: Theorem: The outer measure satisfies: • If A ∈ Σ, then μ∗(A)=μ(A). De ne I k = [k;k+1], to be a countable collection of disjoint bounded intervals that decompose R. Countable Sub-additivity 3. 2.2.

Measure of Open Sets (Approximate from within by Polygons) Measure of Compact Sets (Approximate from outside by Opens) Outer and Inner Measures : 7: Definition of Lebesgue Measurable for Sets with Finite Outer Measure Remove Restriction of Finite Outer Measure (R^n, L, Lambda) is a Measure Space, i.e., L is a Sigma-algebra, and Lambda is a Measure Examples of disjoint sets A and B for which µ∗(A ∪ B) 6= µ∗(A) + µ∗(B) seem at first a bit bizarre.Such an example is given below. Let U Rn be an open set and let F (t;x) a di⁄erentiable function of I U, where I R is an open neighbourhood of 0. outer measure μ∗(A)=inf ∞ i=1 μ(Ai) . Problem 8 (Chapter 2, Q.14). Set G is called the measurable cover of E. In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space.For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume.In general, it is also called n-dimensional volume, n-volume, or simply volume. LEBESGUE MEASURE, INTEGRAL, MEASURE THEORY: A QUICK INTRO 3 The measure generated by balls is the Lebesgue measure. lebesgue measure • page two That is, every subset of R has Lebesgue outer measure which satisfies properties (1)–(3), but satisfies only part of property (4). 4. m∗ is defined on P(R). As stated by Stein and Shakarchi [4], the properties of m (E) are: 1. For instance in R2 we de ne (B r) = area(B r) = ˇr2 for every ball B r 2R2 of radius r. Let’s go into more detail for R; here, the balls are the intervals and we Let I n= [n;n+ 1] for each n2Z, then E= [n2ZE\I n. We have by countable subadditivity that 0

2. m∗ is translation invariant (Proposition 2.2). Problem 14: Show that if a set Ehas positive outer measure, then there is a bounded subset of Ethat also has positive outer measure. So m∗ is close to satisfying the desired four properties. Let us try to compute the derivative d dt t=s Z ’ t (U) F (t;x)dx: Applying the change of variables formula, we have

Lebesgue Outer Measure 3 Note. First we should be a little more precise about our question. Lebesgue outer measure has the following properties: (a) If E 1 E 2, then (E 1) (E 2): (b) The Lebesgue outer measure of any countable set is zero. Approximation of m(E) by m(O) EˆOopen 4. 2 Lecture: Invariant Measures 1. (c) The Lebesgue outer measure of the empty set is zero. For any interval I, m∗(I) = `(I) (Proposition 2.1).