By William Feller

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Let (Ω, F , µ) be a measure space. Let (An )n≥1 be a sequence of elements of F such that An ⊆ An+1 for all n ≥ 1, and let A = ∪+∞ n=1 An (we write An ↑ A). Define B1 = A1 and for all n ≥ 1, Bn+1 = An+1 \ An . 1. Show that (Bn ) is a sequence of pairwise disjoint elements of F such that A = +∞ n=1 Bn . 2. Given N ≥ 1 show that AN = N n=1 Bn . 3. Show that µ(AN ) → µ(A) as N → +∞ 4. Show that µ(An ) ≤ µ(An+1 ) for all n ≥ 1. Theorem 7 Let (Ω, F , µ) be a measure space. Then if (An )n≥1 is a sequence of elements of F , such that An ↑ A, we have µ(An ) ↑ µ(A)1 .

Show that d is a metric on R. 2. Show that if U ∈ TR ¯ , then φ(U ) is open in [−1, 1] 3. Show that for all U ∈ TR ¯ and y ∈ φ(U ), there exists that: ∀z ∈ [−1, 1] , |z − y| < ⇒ z ∈ φ(U ) > 0 such d 4. Show that TR ¯ ⊆ TR ¯. d 5. Show that for all U ∈ TR ¯ and x ∈ U , there is ¯ , |φ(x) − φ(y)| < ∀y ∈ R > 0 such that: ⇒ y∈U d 6. Show that for all U ∈ TR ¯ , φ(U ) is open in [−1, 1]. Tutorial 4: Measurability 10 d 7. Show that TR ¯ ¯ ⊆ TR 8. Prove the following theorem. ¯ TR Theorem 13 The topological space (R, ¯ ) is metrizable.

Then if (An )n≥1 is a sequence of elements of F , such that An ↑ A, we have µ(An ) ↑ µ(A)1 . e. the sequence (µ(An ))n≥1 is non-decreasing and converges to µ(A). Let (Ω, F , µ) be a measure space. Let (An )n≥1 be a sequence of elements of F such that An+1 ⊆ An for all n ≥ 1, and let A = ∩+∞ n=1 An (we write An ↓ A). We assume that µ(A1 ) < +∞. 1. Define Bn = A1 \ An and show that Bn ∈ F, Bn ↑ A1 \ A. 2. Show that µ(Bn ) ↑ µ(A1 \ A) 3. Show that µ(An ) = µ(A1 ) − µ(A1 \ An ) 4. Show that µ(A) = µ(A1 ) − µ(A1 \ A) 5.

### An introduction to probability theory and its applications by William Feller

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