Week 3 Coursework Test

Put $X = {0, 1}^{N}$ . For each finite subset $I$ of $N$ and every function $ϵ : I \to {0, 1}$ define $C (I, ϵ) = ⋂_{i \in I} {x \in X : x (i) = ϵ (i)}$ and call any such subset of $X$ a cylinder set.
1. Thinking of $X$ as the sample space for an infinite sequence of fair coin-tosses, what value in $[0, 1]$ would you consider appropriate as the probability of $C (I, ϵ)$ ?
2. Use your answer to i. to define an outer measure $Ξ$ on $P (X)$ using cylinder sets. Verify that you have indeed defined an outer measure.
3. It is proved in the notes that the collection of subsets of $R$ satisfying the Carathéodory criterion for $Λ$ is a σ-algebra and that $Λ$ is a measure on that σ-algebra. Does that proof need to be changed in order to show that the collection $B$ of subsets of $X$ satisfying the Carathéodory criterion for your $Ξ$ is a σ-algebra and that $Ξ$ is a measure on $B$ ?
4. Verify that every cylinder set belongs to $B$ and that $Ξ (C (I, ϵ))$ agrees with your answer to i.
Fix $f : R \to [0, \infty)$ continuous and non-decreasing. Prove for every $a < b$ that $\int 1_{[a, b]} \cdot f d λ$ is equal to the Riemann integral of $f$ on $[a, b]$ .