Some Notes on Measure Theory Chris Preston This version: February 2005 These notes present the material on measures and kernels which are needed in order to read my lecture notes Specifications and their Gibbs states [16]. They could perhaps be used as a general introduction to some parts of measure theory, but the account is somewhat biased and the contents are determined entirely by the kind of results used in [16]. In particular, the topological aspects of measure theory are missing completely. The theory in [16] really only requires an integral for non-negative mappings, and so we also make this restriction here. The disadvantage is then that we have non-negative cones of mappings instead of vector spaces, but the advantage is that the value 1 is much less of a nuisance and in most cases can be regarded as just another number. There are many texts providing a more balanced account of measure theory. The classical text is Halmos [8] and a very good modern book is Cohn [3]; the first course I gave on the subject was based on Taylor [17]. However, the book everyone should look at at least once is Meyer [14]. Chris Preston Pankow February, 2005 Contents 1 Extended real numbers 4 2 Measurable spaces and mappings 7 3 Measures 18 4 The Carath´eodory extension theorem 24 5 Measures on the real line 30 6 How the integral will be introduced 34 7 Partially ordered sets 35 8 Real valued mappings 40 9 Real valued measurable mappings 47 10 The integral 52 11 The Daniell integral 61 12 The Radon-Nikodym theorem 70 13 Image and pre-image measures 76 14 Kernels 82 15 Product measures 88 16 Countably generated measurable spaces 99 17 The Dunford-Pettis theorem 108 18 Substandard Borel spaces 112 19 The Kolmogorov extension property 116 20 Convergence of conditional expectations 122 21 Existence of conditional distributions 129 3 22 Standard Borel spaces 132 23 The usual diagonal argument 137 Bibliography 139 Index 140 1 Extended real numbers + Most of the mappings we will be dealing with take their values in the set R1 of non-negative extended real numbers. In this chapter we list (without proofs) the facts which we need about these numbers. The reader should check through the list to see that there are no surprises. The natural numbers f0; 1; 2; : : :g are denoted by N, the positive natural numbers f1; 2; : : :g by N+. A countable set is one which is either finite or countably infinite (the latter meaning it has the same cardinality as N). + + + Put R = fa 2 R : a ≥ 0g and let R1 = R [ f1g; the operations of addition + + and multiplication on R will be extended to R1 by letting a + 1 = 1 + a = 1 + + for all a 2 R1, a · 1 = 1 · a = 1 for all a 2 R1 n f0g and 0 · 1 = 1 · 0 = 0. As can be easily verified, these extended operations satisfy the usual associative, commutative and distributive laws (without any restriction). + If a; b 2 R1 then a − b is not always defined; however, it is useful to always assign ja − bj a value: This is the usual value if a; b 2 R+ and 1 in all other + cases, which means that j1 − 1j = 1. Let a; b 2 R1 with a ≤ b; then there + exists c 2 R1 with b = a + c, and c is unique unless both a and b are equal to 1. It is convenient to define b − a to be the `real' difference if a; b 2 R+ and to be + 1 otherwise. Thus b = a + (b − a) for all a; b 2 R1 with a ≤ b and b − a = 1 whenever b = 1. + + The usual total order ≤ on R will be extended to a total order on R1 (also + denoted by ≤) by letting a ≤ 1 for each a 2 R1. In particular a ≤ ja − bj + b + and b ≤ ja − bj + a for all a; b 2 R1. Put a ^ b = minfa; bg, a _ b = maxfa; bg + and note that ja − bj = (a _ b − a) + (a _ b − b) for all a; b 2 R1. We use a ≥ b as an alternative notation for b ≤ a. Moreover, a < b means of course a ≤ b but not a = b, and a > b means a ≥ b but not a = b. + + A sequence fangn≥1 from R1 converges to a 2 R if for each " > 0 there exists m ≥ 1 such that ja − anj < " for all n ≥ m. This means exactly that there exists + p ≥ 1 such that an 2 R for all n ≥ p and that the sequence fangn≥p converges + + to a in R. A sequence fangn≥1 from R1 converges to 1 if for each b 2 R there + exists m ≥ 1 such that an ≥ b for all n ≥ m. If fangn≥1 converges to a 2 R1 then a is uniquely determined by fangn≥1; this value a is called the limit of the sequence and will be denoted by limn!1 an, or mostly just by limn an. If fangn≥1 + is a sequence from R1 then the statement limn an = a is short for the statement that the sequence fangn≥1 converges with limit a. + + Let fangn≥1 be a convergent sequence from R1 with a = limn an. If a 2 R then limn ja − anj = 0. However, if a = 1 then ja − anj = 1 for all n ≥ 1. + A sequence fangn≥1 from R1 is said to be increasing if an ≤ an+1 and decreasing if an+1 ≤ an for all n ≥ 1. Each increasing sequence fangn≥1 converges: If there 4 1 Extended real numbers 5 + exists a 2 R with an ≤ a for all n ≥ 1 then limn an is just the limit in R, and + if no such a 2 R exists then limn an = 1. Moreover each decreasing sequence fangn≥1 also converges: Either an = 1 for all n ≥ 1, in which case limn an = 1, or an < 1 for all n ≥ p for some p ≥ 1, and then limn an is just the limit of the sequence fangn≥p in R. + + Let A be a subset of R1; an element a 2 R1 is said to be an upper bound (resp. lower bound) for A if b ≤ a (resp. b ≥ a) for all b 2 A. An upper bound (resp. lower bound) a is called a least upper bound (resp. greatest lower bound) for A if a ≤ b for each upper bound b for A (resp. if a ≥ b for each lower bound b for A). If a least upper bound (resp. greatest lower bound) exists then it is clearly unique and it will be denoted by sup(A) (resp. by inf(A)). + Each non-empty subset A of R1 possesses both a least upper bound and a greatest lower bound. If A is a bounded subset of R+ (i.e., if b is an upper bound for A for some b 2 R+) then sup(A) is the least upper bound of A in R. If A has no upper bound in R+ then sup(A) = 1. If A = f1g then inf(A) = 1. If A =6 f1g then inf(A) is the greatest lower bound of A n f1g in R. + If fangn≥1 is any sequence from R1 and m ≥ 1 then fan : n ≥ mg will be used + to denote the set of values fa 2 R1 : a = an for some n ≥ mg. + If fangn≥1 is an increasing sequence from R1 then limn an is just the least upper bound of its set of values fan : n ≥ 1g. Thus limn an is the unique element a of + + R1 with the properties: (i) an ≤ a for all n ≥ 1 and (ii) if b 2 R1 with b ≤ a and b =6 a then b ≤ an for all large enough n. Similarly, if fangn≥1 is a decreasing + sequence from R1 then limn an is the greatest lower bound of fan : n ≥ 1g, and + therefore limn an is the unique element a of R1 with the properties: (i) an ≥ a + for all n ≥ 1 and (ii) if b 2 R1 with b ≥ a and b =6 a then b ≥ an for all large enough n. + Let fangn≥1 be any sequence from R1 and for n ≥ 1 let bn = supfam : m ≥ ng and cn = inffam : m ≥ ng. Then the sequence fbngn≥1 is decreasing and fcngn≥1 is increasing. The limits limn bn and limn cn are denoted by lim supn an and lim infn an respectively. Then lim infn an ≤ lim supn an, with equality if and only if the sequence fangn≥1 converges. Moreover, if this is the case then lim inf an = lim an = lim sup an : n!1 n!1 n!1 + + + 0 0 A binary operation ? : R1 × R1 ! R1 is said to be monotone if a ? b ≤ a ? b 0 0 whenever a ≤ a and b ≤ b . If ? is monotone and fangn≥1 and fbngn≥1 are + increasing sequences from R1 then fan ? bngn≥1 is also an increasing sequence. A monotone operation ? is defined to be continuous if lim (an ? bn) = lim an ? lim bn n!1 n!1 n!1 1 Extended real numbers 6 + holds for all increasing sequences fangn≥1 and fbngn≥1 from R1. + The operations +, ·, _ and ^ on R1 are all continuous (and so in particular monotone). Moreover, for all a; b 2 R+ the operation (c; d) 7! ac + bd is also continuous. Furthermore, these operation are all finite, where a binary operation + + + ? on R1 is said to be finite if a ? b 2 R for all a; b 2 R .