Chapter Questions
Prove that every monotone function is measurable.
Prove that if $f$ is a measurable function, then the level set $\{x: f(x)=a\}$. is measurable for every $a \in \overline{\mathbb{R}}$.
Show that if $f$ is measurable, then the truncation of $f$ :$$f^{a}(x)= \begin{cases}a & \text { if } f(x)>a \\ f(x) & \text { if } f(x) \leq a\end{cases}$$is also measurable.
Find a non-measurable $f$ such that $f^{2}$ is measurable.Passage to the limit does not destroy measurability - all the work needed was done when we established the stability properties of $\mathcal{M} !$
Let $f_{n}$ be a sequence of measurable functions. Show that the set $E=$ $\left\{x: f_{n}(x)\right.$ converges $\}$ is measurable.
Show that for measurable $f$, ess sup $f \leq \sup f$. Show that these quantities coincide when $f$ is continuous.
Show that $\mathcal{F}_{X}$ is the smallest $\sigma$-field containing the inverse images $X^{-1}(B)$ of all Borel sets $B$.
Is the family of sets $\{X(A): A \in \mathcal{F}\}$ a $\sigma$-field?
Find the function $f$ for a down-and-out call (which is a European call except that is ceases to exist if the stock price at any time before the exercise date goes below the barrier $L<S(0))$.