# An introduction to probability theory by Geiss C., Geiss S.

By Geiss C., Geiss S.

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3 CHAPTER 3. INTEGRATION Connections to the Riemann-integral In two typical situations we formulate (without proof) how our expected value connects to the Riemann-integral. 4. 1 Let f : [0, 1] → ❘ be a continuous function. Then 1 0 f (x)dx = ❊f with the Riemann-integral on the left-hand side and the expectation of the random variable f with respect to the probability space ([0, 1], B([0, 1]), λ), where λ is the Lebesgue measure, on the right-hand side. 14). The function p is called density of the measure P.

We close with a simple deviation inequality for f . 9 Let f be a random variable defined on a probability space (Ω, F, P) such that ❊f 2 < ∞. Then one has, for all λ > 0, ❊ (f − ❊f )2 ❊ f2 P(|f − ❊f | ≥ λ) ≤ ≤ 2 . λ2 λ Proof. 1 to |f − ❊f |2 gives that ❊f exists. Ap- P({|f − ❊f | ≥ λ}) = P({|f − ❊f |2 ≥ λ2}) ≤ ❊|f −λ2❊f | 2 Finally, we use that ❊(f − ❊f )2 = ❊f 2 − (❊f )2 ≤ ❊f 2. 62 CHAPTER 3. 1 Definitions Let us introduce some basic types of convergence. 1 [Types of convergence] Let (Ω, F, P) be a probability space and f, f1 , f2 , · · · : Ω → ❘ be random variables.

2. As π-system I we take the system of all F = A×B with A ∈ F1 and B ∈ F2 . Letting f (ω1 , ω2 ) = 1IA (ω1 )1IB (ω2 ) we easily can check that f ∈ H. For instance, property (c) follows from Ω1 ×Ω2 f (ω1 , ω2 )d(P1 × P2 ) = (P1 × P2 )(A × B) = P1 (A)P2 (B) and, for example, Ω1 Ω2 f (ω1 , ω2 )dP2 (ω2 ) dP1 (ω1 ) = = Ω1 1IA (ω1 )P2 (B)dP1 (ω1 ) P1(A)P2(B). 2 gives that H consists of all bounded functions f : Ω1 × Ω2 → ❘ measurable with respect F1 × F2 . Hence we are done. Now we state Fubini’s Theorem for general random variables f : Ω1 × Ω2 → ❘.