I was trying to solve an exercise in my book (Cinlar - Probability and Stochastics - II.1.20) but I have some problems understanding what it asks of me. The exercise asks for the following.
Ex. Let $c:\mathbb R\to [0,1]$ a cumulative distribution function, that is increasing and right continuous. Let $q:(0,1)\to \mathbb R$ the corresponding quantile function (i.e. $q$ is the generalized inverse of $c$; $q(u)=inf \{t|c(t)>u\}$). Let $L$ denote the Lebesgue measure on $(0,1)$ and put $\mu=L \circ q^{-1}$.
(a) Show that $\mu$ is a measure on $\overline{ \mathbb R} $.
(b) Show tha $\mu$ is the distribution on $\overline{ \mathbb R} $ corresponding to the distribution funcion $c$. Thus, to every distribution $c$ on $\mathbb R$ there corresponds an unique probability measure $\mu $ on $\overline{ \mathbb R} $ and viceversa.
My doubts (only in the b point):
I didn't understand what the author means by "distribution on $\overline{ \mathbb R} $ corresponding to the distribution funcion $c$". In fact the only type of distribution defined in the book is the measure $\mu_X:\mathcal B_{\mathbb R}\to [0,1]$ relative to a random variable $X:\Omega \to \mathbb R$ (that is $\mu _X(A)=\mathbb P(X^{-1}(A))$). What does he mean? That exists a random variable $X:\Omega \to \mathbb R$ such that $c_{X}=c$ ?
NB: by definition the author put $c_{X}(y)=\mathbb P(X^{-1}]-\infty,y])$ for all $y \in\mathbb R$.
Part (a) is a change fo variables. You have that $q$ is a map from the measure space $((0,1),\mathscr{B}(0,1)),L)$ to the measurable space $(\mathbb{R},\mathscr{B})$. You can convince yourself that the map $\mu:\mathscr{B}(\mathbb{R})\rightarrow[0,1]$ given by $\mu(B)=L(q^{-1}(B))$ is indeed a measure (in fact a probability measure on $(\mathbb{R},\mathbb{B}(\mathbb{R}))$.
The second part requires you to convince yourself that $$c(x)\geq t\quad\text{if and only if}\quad q(t)\leq x$$
So $\mu((-\infty,x])=L(\{t:q(t)\leq x\})=L(\{t:c(x)\geq t\})=c(x)$