Probability with Measure

3.7 Exercises on Chapter 3

3.1 Show that the following functions are Borel measurable, as maps from \(\R \) to itself.
- (a) The constant function \(f(x)=\alpha \), where \(\alpha \in \R \).
- (b) \(g(x)=\begin {cases} 0 & \text { for }x < 0 \\ e^x & \text { for } x \geq 0. \end {cases}\)
- (c) \(h(x)=\sin (\cos (x))\)
- (d) \(i(x)=\sin (x^2\1_{[0,\infty )}(x))\)
3.2 Let \((S, \Sigma )\) be a measurable space and let \(f:S \rightarrow \R \). Show that \(f\) is measurable if and only if \(f^{-1}((a,b)) \in \Sigma \) for all \(-\infty \leq a < b \leq \infty \).
3.3 Let \((S, \Sigma )\) be a measurable space and let \(f:S \rightarrow \R \) be measurable. Show that \(|f|\) is measurable.
3.4 Let \(f:\R \to \R \) and let \(\alpha \in \R \).
- (a) Suppose that \(f\) is Borel measurable. Show that the mapping \(h:\R \rightarrow \R \) is measurable, where \(h(x) = f(x+\alpha )\).
- (b) Suppose that \(f\) is differentiable. Explain why both \(f\) and its derivative \(f^{\prime }\) are measurable functions.
- (c) Suppose that \(f\) is monotone increasing. Show that \(f\) is measurable.
  
  Hint: Show that \(f^{-1}((c,\infty ))\) is an interval. Recall that \(I\sw \R \) is an (open, closed or half-open) interval of \(\R \) if, whenever \(a,b\in I\) and \(a<c<b\) we have \(c\in I\).
3.5 Let \((S,\Sigma )\) be a measurable space. Show that the set \(V\) of simple functions \(f:S\to \R \), with pointwise addition and scalar multiplication, is a real vector space.
3.6 \((\Delta )\) Let \((S, \Sigma )\) be a measurable space and \(f:S\to \R \) be measurable. Let \(\alpha \in \R \).

In this question you may not use the algebra of measurable functions (Theorem 3.1.5) or any of the results in Section 3.4. We use the results (a)-(c) in the proof of Theorem 3.1.5.
- (a) Show that \(g = f + \alpha \) is measurable.
- (b) Show that \(g = \alpha f\) is measurable.
- (c) Let \(G:\R \to \R \) be Borel measurable. Show that \(G\circ f:S\to \R \) is measurable, where \((G\circ f)(x)=G(f(x))\).
3.7 \((\Delta )\) Recall the definition of an open set, from Definition 3.3.1.
- (a) Let \(O_1\) and \(O_2\) be open subsets of \(\R \). Show that \(O_1\cup O_2\) and \(O_1\cap O_2\) are also open.
- (b) For each \(n\in \N \) let \(O_n\) be an open subset of \(\R \). Consider the following claims:
  - (i) \(A=\bigcup _{n\in \N } O_n\) is open.
  - (ii) \(B=\bigcap _{n\in \N } O_n\) is open.
  Which of these claims are true? Give a proof or a counterexample in each case.
- (c) A set \(C\sw \R \) is said to be closed if \(\R \sc C\) is open. Which of your results from parts (a) and (b) hold for closed sets?

Challenge questions

3.8 A function \(f:\R \rightarrow \R \) is said to be upper-semicontinuous at \(x \in \R \), if given any \(\eps > 0\) there exists \(\de > 0\) so that \(f(y) < f(x) + \eps \) whenever \(|x -y| < \de \).
- (a) Show that \(f = {\1}_{[a, \infty )}\) (where \(a \in \R \)) is upper-semicontinuous for all \(x \in \R \),
- (b) Deduce that the floor function \(f(x) = \lfloor x \rfloor \), which is equal to the greatest integer less than or equal to \(x\), is upper-semicontinuous at all \(x \in \R \).
- (c) Show that if \(f\) is upper-semicontinuous for all \(x \in \R \) then \(f\) is measurable.