\[ \newcommand{\C}{\mathbb{C}} \newcommand{\haar}{\mathsf{m}} \newcommand{\B}{\mathscr{B}} \newcommand{\D}{\mathscr{D}} \newcommand{\P}{\mathcal{P}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\g}{>} \newcommand{\l}{<} \newcommand{\intd}{\,\mathsf{d}} \newcommand{\Re}{\mathsf{Re}} \newcommand{\area}{\mathop{\mathsf{Area}}} \newcommand{\met}{\mathop{\mathsf{d}}} \newcommand{\emptyset}{\varnothing} \DeclareMathOperator{\borel}{\mathsf{Bor}} \DeclareMathOperator{\baire}{\mathsf{Baire}} \newcommand{\symdiff}{\mathop\triangle} \]

1. Put $X = \{0,1\}^\N$. For each finite subset $I$ of $\N$ and every function $\epsilon : I \to \{0,1\}$ define \[ \mathsf{C}(I,\epsilon) = \bigcap_{i \in I} \, \{ x \in X : x(i) = \epsilon(i) \} \] 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 $\mathsf{C}(I,\epsilon)$?
   2. Use your answer to i. to define an outer measure $\Xi$ on $\mathcal{P}(X)$ 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 $\Lambda$ is a σ-algebra and that $\Lambda$ is a measure on that σ-algebra. Does that proof need to be changed in order to show that the collection $\mathscr{B}$ of subsets of $X$ satisfying the Carathéodory criterion for your $\Xi$ is a σ-algebra and that $\Xi$ is a measure on $\mathscr{B}$?
   4. Verify that every cylinder set belongs to $\mathscr{B}$ and that $\Xi(\mathsf{C}(I,\epsilon))$ agrees with your answer to i.
2. Fix $f : \R \to [0,\infty)$ continuous and non-decreasing. Prove for every $a \l b$ that \[ \int 1_{[a,b]} \cdot f \intd \lambda \] is equal to the Riemann integral of $f$ on $[a,b]$.