Classes of continuous functions 240 56. Recall that a set of real numbers is open if and only if it is a countable disjoint union of open intervals. Full subgroups 250 58. regular Borel probability measure. Generation of Borel measures 231 54. (Plancherel-Parseval Formula) For any finite Borel measure µ and any bounded,continuousfunction f:R!R withcompactsupport, Z f (x)dµ(x)=lim "!0 1 2º Z R fˆ(µ)µˆ(°µ)e°" 2µ /2dµ. If f : R !R is Borel measurable and g: Rn!R is Lebesgue (or The natu-ral σ-field on which to define a probability measure on the line is the Borel σ-field. Uniqueness 262 CHAPTER XII: MEASURE AND TOPOLOGY IN GROUPS 61. Borel probability measure on B, and Eis measure hyper nite if E is -hyper nite for every Borel probability measure on X. Existence 251 59. We use P(X) to denote the space of all Borel probability measures on X, equipped with the Polish topology generated by the functions of the form 7! Then the event A(i:o:) = fA n ocurrs for in nitely many n gis given by A(i:o:) = \1 k=1 [1 n=k A n; Lemma 1 Suppose that fA n: n 1gis a sequence of events in a probability space. Theorem 7.13. grals and is very useful for the construction of countably additive probability measures on the real line. Measure and probability Peter D. Ho September 26, 2013 This is a very brief introduction to measure theory and measure-theoretic probability, de- Linear functional 243 CHAPTER XI: HAAR MEASURE 57. Setting for this section: We consider (f,µ)wheref: M → M is a C2 dif-feomorphism of a compact Riemannian manifold M and µ is an f-invariant Borel probability measure. 51. Measurable groups 257 60. As a countable Borel equivalence relation is hyper nite if and only if it is Borel reducible to E 0 (see Theorem 1.3.8), it immediately follows that a countable Borel equivalence relation is invariant-measure hyper nite Regular measures 223 53. w.r.t.

• To define a probability measure over a Borel field, we first assign probabilities to the intervals in a consistent way, i.e., in a way that satisfies the axioms of probability For example to define uniform probability measure over (0,1), we first assign P((a,b)) = b−a to all intervals Borel’s Paradox. (7.15) The hypothesis that f have compact support is needed to guarantee that the Fourier transform fˆ is well-defined.The factor e°"2µ2/2 in the integral is needed because in Borel-Cantelli Lemmas Suppose that fA n: n 1gis a sequence of events in a probability space. If R fdν = 1 for an f ≥ 0 a.e. 1.4. The convolution of a probability measure &on with a Borel probability measure on Xis the Borel probability measure & on Xgiven by & = R d&(). We say that is &-stationary if = & .

ν, then λ is a probability measure and f is called its probability density function (p.d.f.) to Borel measurable functions are Lebesgue measurable. Consequence: If f is Borel on (Ω,F) and R A fdν = 0 for any A ∈ F, then f = 0 a.e. Borel Sets 1 Chapter 1. Open Sets, Closed Sets, and Borel Sets Section 1.4. Consider a sphere equipped with lines of latitude (red) and longitude (blue): Suppose we take a point at random from a uniform distribution over the surface of that sphere (i.e., a distribution that makes the probability that the point lies within a particular region proportional to that region’s area). The push-forward of a measure on Xvia a function f: X!Y is the mea-sure f on Y given by f (B) = (f 1(B)). Bof R, and it is Borel measurable if f 1(B) is a Borel measurable subset of Rn for every Borel subset Bof R This de nition ensures that continuous functions f: Rn!R are Borel measur-able and functions that are equal a.e. Regular contents 237 55. 2 Entropy, Lyapunov Exponents and Dimension In this section, we focus on a set of ideas in which entropy plays a central role. For any probability measure P on (Rk,Bk) corresponding A Borel probability measure on X is a Borel measure on X for which (X) = 1. ν. Borel Sets Note. disjoint Borel subsets of X. The proof will again be only sketched. Also recall that: 1. a countable union of open sets is open, and 2. a countable intersection of closed sets is closed. Borel sets and Baire sets 219 52. If X1 n=1 P(A n) < 1; (1)