Chapter Questions
Show that we get an equivalent notion if in the above definition we replace the word 'intervals' by any of these: 'open intervals', 'closed intervals', 'the intervals of the form $(a, b]$, 'the intervals of the form $[a, b)$ '.
Prove that $C$ is uncountable.
Prove that the Lebesgue function $F$ is continuous and sketch its graph.
The outer measure of an interval equals its length.
Prove that if $m^{*}(A \Delta B)=0$, then $m^{*}(A)=m^{*}(B)$.
Find a formula describing $m(A \cup B)$ and $m(A \cup B \cup C)$ in terms of measures of the individual sets and their intersections (we do not assume that the sets are pairwise disjoint).
Show that the family of intervals of the form $(a, b]$ also generates the $\sigma$-field of Borel sets. Show that the same is true for the family of all intervals $[a, b)$.
Show that each of the following two statements is equivalent to saying that $E \in \mathcal{M}$ :(i) given $\varepsilon>0$ there is an open set $O \supset E$ with $m^{*}(O \backslash E)<\varepsilon$,(ii) given $\varepsilon>0$ there is a closed set $F \subset E$ with $m^{*}(E \backslash F)<\varepsilon$
Prove that if $H_{i}$ are pairwise disjoint events such that $\bigcup_{i=1}^{\infty} H_{i}=\Omega$ $P\left(H_{i}\right) \neq 0$, then$$P(A)=\sum_{i=1}^{\infty} P\left(A \mid H_{i}\right) P\left(H_{i}\right)$$
Suppose that $A$ and $B$ are independent events. Show that $A^{c}$ and $B$ are also independent.
Suppose $N=5, U=1.2, D=0.9$, and $S(0)=500$. Find the number of all paths. How many paths lead to the price $S(5)=524.88 ?$ What is the probability that $S(5)>900$ if the probability going up in a single step is $0.5 ?$
Prove that $\mathcal{F}_{m}$ has $2^{2^{m}}$ elements.
Prove that the sequence $\mathcal{F}_{m}$ is increasing.
Prove that $\mathcal{G}_{m}$ and $\mathcal{G}_{k}$ are independent if $m \neq k$.