## Highlights - − additivity (Rene Schilling 17) - area · Aj = areaAj j∈ j∈ (Rene Schilling 17) - Inverse images and set operations are, however, always compatib (Rene Schilling 20) - Set operations and direct images under a map f are not necessarily compatible: (Rene Schilling 20) - X ∈ (Rene Schilling 29) - A ∈ =⇒ Ac ∈ (Rene Schilling 29) - Aj j∈ ⊂ =⇒ Aj ∈ (Rene Schilling 29) - A -algebra (Rene Schilling 29) - A set A ∈ is said to be \(-\)measurable\. (Rene Schilling 29) - \) A B ∈ =⇒ A ∪ B ∈ \. (Rene Schilling 29) - i\) Aj j∈ ⊂ =⇒ j∈ Aj ∈ \. (Rene Schilling 29) - 3\.1 Definition A -algebra on a set X is a family of subsets of X with the following properties: X ∈ A ∈ =⇒ Ac ∈ A j j∈ ⊂ =⇒ Aj ∈ j∈ (Rene Schilling 29) - \(ii\) A B ∈ =⇒ A ∪ B ∈ \. (Rene Schilling 29) - \(iii\) Aj j∈ ⊂ =⇒ j∈ Aj Indeed: if Aj ∈ , then Acj Aj ∈ \. Acj ∈ (Rene Schilling 29) - 3\.4 Theorem \(and Definition\) \(i\) The i -algebras i in X is again a -algebra The intersection i∈I i of arbitrarily many lgebra in X\. \(ii\) For every system of sets ⊂ X there exists a smallest \(also: minimal, coarsest\) -algebra containing \. This is the -algebra generated by , denoted by , and is called its generator\. (Rene Schilling 30) - Th (Rene Schilling 31) - The -algebra n generated by the open sets n of n is (Rene Schilling 31) - called Borel -algebra, a (Rene Schilling 31) - A family of subsets of a general space X satisfying the conditions (Rene Schilling 31) - 1 – 3 is called a topology, (Rene Schilling 31) - 3\.6 Definition The -algebra n generated by the open sets n of n is called Borel -algebra, and its members are the Borel sets or Borel measurable sets\. We write n or n for the Borel sets in n \. (Rene Schilling 31) - A \(positive\) measure on X is a mapping → 0 defined (Rene Schilling 36) - on a -algebra satisfying (Rene Schilling 36) - ∅ = 0 (Rene Schilling 36) - and, for any countable family of pairwise disjoint sets Aj j∈ ⊂ , (Rene Schilling 36) - · Aj = Aj j∈ j∈ (Rene Schilling 36) - An exhausting sequence Aj j∈ ⊂ is an increasing sequence of sets A1 ⊂ (Rene Schilling 36) - A2 ⊂ A3 ⊂ such that j∈ Aj = X\. (Rene Schilling 36) - 4\.1 Definition A \(positive\) measure on X is a mapping → 0 defined on a -algebra satisfying ∅ = 0 \(M1 \) and, for any countable family of pairwise disjoint sets Aj j∈ ⊂ , -additivity · Aj j∈ = j∈ Aj j∈ (Rene Schilling 36) - -additivity (Rene Schilling 36) - \(M 1 \) (Rene Schilling 36) - \(M 2 \) (Rene Schilling 36) - ce, A measure is said to be -finite and ce, if contains an exhausting sequence A j j∈ such that Aj < for all (Rene Schilling 36) - l j ∈ \. (Rene Schilling 36) - 4\.3 Proposition (Rene Schilling 37) - \( finitely additive\) (Rene Schilling 37) - \(monotone\) (Rene Schilling 37) - \(strongly additive\) (Rene Schilling 37) - \(subadditive\) (Rene Schilling 37) - B \ A = B − A (Rene Schilling 37) - A map → 0 is a (Rene Schilling 38) - measure if, and only if, (Rene Schilling 38) - ∅ = 0, (Rene Schilling 38) - A ∪ · B = A + B f (Rene Schilling 38) - for any increasing sequence Aj j∈ ⊂ with Aj ↑ A ∈ we have (Rene Schilling 38) - A = lim Aj = sup Aj j→ j∈ A ∈ , \(iii\) can be replaced by either (Rene Schilling 38) - 4\.4 Theorem Let X measure if, and only if, be (Rene Schilling 38) - \(continuity of measures from below\) (Rene Schilling 38) - \(continuity of measures from above\) (Rene Schilling 38) - \(continuity of measures at ∅\) (Rene Schilling 38) - Proof \(of Theorem 4\.4\) (Rene Schilling 38) - 4\.6 Corollary Every measure [pre-measure] is -subadditive, i\.e (Rene Schilling 40) - Aj Aj j∈ j∈ (Rene Schilling 40) - holds for all sequences Aj j∈ ⊂ of not necessarily disjoint sets (Rene Schilling 40) - \(iii\) follows\. (Rene Schilling 40) - 4\.8 Definition The set-function n on n n that assigns every half-open rectangle a b = a1 b1 × · · · × an bn ∈ the value n n a b = bj − aj j=1 is called n-dimensional Lebesgue measure\. (Rene Schilling 41) - 4\.9 Theorem Lebesgue measure n exists, is a measure on the Borel sets n and is unique\. Moreover, n enjoys the following additional properties for B ∈ n : \(i\) n is invariant under translations: n x + B = n B , x ∈ n ; \(ii\) n is invariant under motions: n R−1 B = n B where R is a motion, i\.e\. a combination of translations, rotations and reflections; \(iii\) n M −1 B = det M−1 n B for any invertible matrix M ∈ n×n \. (Rene Schilling 42) - Let 1 2 be countably many measures on X and let j j∈ be a sequence of positive numbers\. Show that A = j=1 j j A , A ∈ , is again a measure\. [Hint: to show -additivity use \(and prove\) the following helpful lemma: for any double sequence ij i j ∈ , of real numbers we have sup sup ij = sup sup ij (Rene Schilling 42) - Prove that is a Borel set and that = 0 in two ways: (Rene Schilling 43) - We have seen in Problem 4\.10 that measurable subsets of null sets are again null sets (Rene Schilling 43) - a measure space X ∗ \(or a measure \) is complete if all subsets of -null sets are again in ∗ (Rene Schilling 43) - In other words: if all subsets of a null set are null sets\. (Rene Schilling 43) - 5\.8 Theorem \(i\) n-dimensional Lebesgue measure n is invariant under translations, i\.e\. n n x + B = B ∀ x ∈ n ∀ B ∈ n \(5\.3\) \(ii\) Every measure on n n which is invariant under translations and satisfies = 0 1n < is a multiple of Lebesgue measure: = n \. (Rene Schilling 48) - Proof (Rene Schilling 48) - Dilations\. Mimic the proof of Theorem 5\.8\(I\) and show that t · B = tb b ∈ B is a Borel set for all B ∈ n and t > 0\. Moreover, n t · B = tn n B ∀ B ∈ n ∀ t > 0 (Rene Schilling 50) - Construct an open and unbounded set in with finite, strictly positive Lebesgue measure\. (Rene Schilling 61) - Construct an open, unbounded and connected set in 2 with finite, strictly positive Lebesgue measure\. (Rene Schilling 61) - Is there a connected, open and unbounded set in with finite, strictly positive Lebesgue measure? (Rene Schilling 61) - Borel–Cantelli lemma \(1\) – the direct half\. Prove the following theorem\. Theorem \(Borel–Cantelli lemma\)\. Let P be a probability space\. For every sequence Aj j∈ ⊂ we have PAj < =⇒ P Aj = 0 \(6\.11\) j=1 n=1 j=n [Hint: use Theorem 4\.4 and the fact that P jn Aj jn PAj \.] Remark\. This is the ‘easy’ or direct half of the so-called Borel–Cante Remark\. This is the ‘easy’ or direct half of the so-called Borel–Cantelli lemma; the more difficult part see T18\.9 (Rene Schilling 61) - The condition ∈ Aj means that happens n=1 j=n to be in infinitely many of the Aj and the lemma gives a simple sufficient condition when certain events happen almost surely not infinitely often, i\.e\. only finitely often with probability one\. (Rene Schilling 61) - 7\.1 Definition Let X X ′ ′ be two measurable spaces\. A map T X → X ′ is called /′ -measurable \(or measurable unless this is too ambiguous\) if the pre-image of every measurable set is a measurable set: T −1 A′ ∈ ∀ A′ ∈ ′ \(7\.1\) A random variable is a measurable map from a probability space to any measurable space (Rene Schilling 63) - The following lemma shows that measurability needs only to be checked for the sets of a generator\. 7\.2 Lemma Let X X ′ ′ be measurable spaces and let ′ = ′ \. Then T X → X ′ is /′ -measurable if, and only if, T −1 ′ ⊂ , i\.e\. if T −1 G′ ∈ ∀ G′ ∈ ′ (Rene Schilling 64) - Proof (Rene Schilling 64) - continuous maps are measurable (Rene Schilling 64) - From calculus1 we know that T is continuous if, and only if, T −1 U ⊂ n is open ∀ open U ⊂ m \(7\.5\) Since the open sets m in m generate the Borel -algebra m , we can use \(7\.5\) to deduce T −1 m ⊂ n ⊂ n = n By Lemma 7\.2, T −1 m ⊂ n which means that T is measurable\. (Rene Schilling 64) - Not every measurable map is continuous, e\.g\. x → 1−11 x\. (Rene Schilling 64) - 7\.5 Definition \(and Lemma\) Let Ti i∈I be arbitrarily many mappings Ti X → X i from the same space X into measurable spaces Xi i \. The smallest -algebra on X that makes all Ti simultaneously measurable[] is T i i ∈ I = Ti−1 i \(7\.6\) i∈I We say that Ti i ∈ I is generated by the family Ti i∈I \. (Rene Schilling 65) - On is the set of all orthogonal n × n matrices: T ∈ On if, and only if, t T · T = id\. (Rene Schilling 66) - 7\.9 Theorem If T ∈ On, then n = T n \. (Rene Schilling 66) - Tx − Ty = Tx − y = x − y (Rene Schilling 66) - hence measurable by Example 7\.3\. (Rene Schilling 66) - 7\.10 Theorem Let S ∈ GLn \. Then 1 n n n S = det S −1 = \(7\.9\) det S (Rene Schilling 67) - Proof (Rene Schilling 67) - 7\.11 Corollary Lebesgue measure is invariant under motions: n = M n for all motions M in n \. In particular, congruent sets have the same measure\. (Rene Schilling 68) - Proof We know that M is of the form x T \. Since det T = ±1, we get n n 7\.10 n 5\.8 n n M = x T = x = n (Rene Schilling 68) - Remark\. Sets with empty interior are called nowhere dense (Rene Schilling 69) - Cantor’s ternary set\. E0 = 0 1\. Remove th (Rene Schilling 69) - u −1 a = x ∈ X ux ∈ a = x ∈ X ux a ∈ (Rene Schilling 71) - 8\.3 Lemma ¯ ¯ is generated by all sets of the form a \(or b or − c or − d\) where a \(or b c d\) is from or \. (Rene Schilling 72) - 8\.6 Definition A simple function g X → on a measurable space X is a function of the form \(8\.6\) with finitely many sets A1 AM ∈ and y 1 yM ∈ \. The set of simple functions is denoted by or \. (Rene Schilling 74) - If the sets Aj 1 j M, are mutually disjoint we call M y j 1Aj x j=0 with y0 = 0 and A0 = A1 ∪ ∪ AM c a standard representation of g\. (Rene Schilling 74) - If a measurable function h X → attains only finitely many values y 1 y2 yM ∈ , then it is a simple function\. (Rene Schilling 74) - ¯ 8\.9 Corollary Let X be a measurable space\. If uj X → , ¯ j ∈ , are , measurable functions, then so are sup uj inf uj lim sup uj lim inf uj j∈ j∈ j→ j→ and, whenever it exists, limj→ uj \. (Rene Schilling 76) - Proof \(of Corollary 8\.9\) (Rene Schilling 77) - ¯ 8\.10 Corollary Let u v be /-measurable ¯ -measurable numerical functions\. Then the functions u ± v uv u ∨ v = max u v u ∧ v = min u v \(8\.14\) ¯ are /-measurable ¯ -measurable \(whenever they are defined\)\. (Rene Schilling 77) - Proof \(of Corollary 8\.10\) (Rene Schilling 78) - ¯ 8\.12 Corollary If u v are /-measurable ¯ -measurable numerical functions, then u