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By Geiss C., Geiss S.

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2) x + (y + z) = (x + y) + z form all x, y, z ∈ L. (3) There exists an 0 ∈ L such that x + 0 = x for all x ∈ L. (4) For all x ∈ L there exists an −x such that x + (−x) = 0. (5) 1 x = x. (6) α(βx) = (αβ)x for all α, β ∈ ❘ and x ∈ L. 52 CHAPTER 3. INTEGRATION (7) (α + β)x = αx + βx for all α, β ∈ ❘ and x ∈ L. (8) α(x + y) = αx + αy for all α ∈ ❘ and x, y ∈ L. Usually one uses the notation x − y := x + (−y) and −x + y := (−x) + y etc. Now we state the Monotone Class Theorem. It is a powerful tool by which, for example, measurability assertions can be proved.

36 CHAPTER 2. 2 [independence of a finite family of random variables] Let (Ω, F, P) be a probability space and fi : Ω → ❘, i = 1, . . , n, random variables. The random variables f1 , . . , fn ∈ Bn) = P (f1 ∈ B1) · · · P (fn ∈ Bn) . 9 what does it mean that a sequence of events is independent. Now we rephrase this definition for arbitrary families. 3 [independence of a family of events] Let (Ω, F, P) be a probability space and I be a non-empty index-set. , one has that P (Ai1 ∩ · · · ∩ Ain ) = P (Ai1 ) · · · P (Ain ) .

S. s. and ❊|f g| = 0. Hence we may set f˜ := f (❊|f |p ) 1 p and g˜ := g (❊|g|q ) q 1 . We notice that xa y b ≤ ax + by for x, y ≥ 0 and positive a, b with a + b = 1, which follows from the concavity of the logarithm (we can assume for a moment that x, y > 0) ln(ax + by) ≥ a ln x + b ln y = ln xa + ln y b = ln xa y b . Setting x := |f˜|p , y := |˜ g |q , a := p1 , and b := 1q , we get 1 1 q g| |f˜g˜| = xa y b ≤ ax + by = |f˜|p + |˜ p q 60 CHAPTER 3. INTEGRATION and ❊|f˜g˜| ≤ p1 ❊|f˜|p + 1q ❊|˜g|q = p1 + 1q = 1.

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