My hope in asking for an alternative proof to the following theorem is to sharpen my intuition.
Change of variable theorem:
Let $h$ be Borel-measurable and $X, g(X) \in \mathscr L^1(\Omega, \mathscr F, \mathbb P)$
$$E[h(X)] := \int_{\Omega} h(X) d \mathbb P = \int_{\mathbb R} h(t) d\mathcal L_X(t)$$
Apparently, this is proved using standard machine, i.e. the definition of Lebesgue integration, i.e. with 4 steps:
- Assume $X$ is an indicator function $1_A$
- Assume $X$ is a nonnegative simple function $\sum_i 1_{A_i}$
- Assume $X$ is a nonnegative function $\sup \sum_i 1_{A_i}$
- Assume $X$ is an integrable function. $f^+ - f^-$
Now I didn't understand this at first because I took this as the definition of expectation and actually didn't understand $\int_{\Omega} h(X) d \mathbb P$. After re-discovering Skorokhod representations (*), I think I finally do.
Instead of using standard machine, why don't we just say that change of variable theorem holds because every random variable $X$ has the identity random variable in $(\mathbb R, \mathscr B(\mathbb R), \mathcal L_X)$ as a Skorokhod representation because of the proposition here:
$$X(\omega) := \sup \{x \in \mathbb R | 1-e^{-x \lambda} < \omega\} \ \text{is a random variable.}$$
?
(*)
Apparently, $((0,1), \mathscr B(0,1), Leb)$, $$X(\omega) := \frac{1}{\lambda} \ln(\frac{1}{1-\omega})$$
has exponential distribution. Actually, $((0,1), \mathscr B(0,1), Leb)$, the identity random variable $$X(\omega) = \omega$$ has uniform distribution while in $(\mathbb R, \mathscr B(\mathbb R), 1-e^{-x \lambda})$ the identity random variable has exponential distribution, well, by definition.
So, the 'exponential distribution' has various forms depending on the probability space, and they're all given by
$$X(\omega) := \sup \{x \in \mathbb R | 1-e^{-x \lambda} < \omega\}$$
Even within the same probability space, an exponential distribution can have various forms such as
$$X(1-\omega) := \frac{1}{\lambda} \ln(\frac{1}{\omega})$$
or more generally with a measure preserving function $g$:
$$X(g(\omega)) := \frac{1}{\lambda} \ln(\frac{1}{1-g(\omega)})$$
Given this, change of variable seems to follow merely and immediately from a change of Skorokhod representation from $$X(\omega) \in L^1(\Omega, \mathscr F, \mathbb P)$$ to $$t \in L^1(\mathbb R, \mathscr B(\mathbb R), \mathcal L_X)$$
In the latter the identity random variable is exponential by definition, construction or tautology, not sure of term.