Normalizing a function with an area Consider a continuously differentiable and strictly decreasing function $G:[0,T]\rightarrow [0,1]$ for some $T>0$ with $G(0)=1,G(T)=0$.
Suppose you normalize $G$ with the area under its graph and define $f(t)$ to be
$$f(t)=\frac{G(t)}{\int^T_0G(x)dx} \text{ if $t\in(0,T)$, 0 otherwise.}$$
My questions:

*

*Why does normalizing G by the entire area under the curve produce something that behaves exactly like a probability density function?


*What is the intuition behind #1?
 A: Any nonnegative function $f$ that integrates to 1 is the pdf of some random variable — namely, the random variable with cdf $F(x) :=\int_{-\infty}^x f(x) dx$.
You simply went straight to defining a density — your normalized $G(x)$ — as opposed to motivating it with some random variable or cdf.
The key step here is that you ensured that the normalized function (let's call it $g(x)$) integrates to 1 over $[0,T]$, hence you can answer questions like $P(X \in [0,0.5T])= \int_0^{0.5T}g(x)dx$
A: Theorem. Let $f$ be a non-negative function (Lebesgue) integrable over some nice (Borel) set $\mathrm{V}$ of $\mathbf{R}^d.$ There exists a probability space $(\Omega, \mathscr{F}, \mathbf{P})$ and a random vector $X:\Omega \to \mathbf{R}^d$ such that $\mathbf{P}(X \in \mathrm{A}) = c \int\limits_\mathrm{A} f,$ where the normalising constant is $c^{-1} = \int\limits_\mathrm{V} f.$
Proof. Let $\Omega = \mathrm{V},$ $\mathscr{F}$ the Borel sets of $\mathrm{V}$ and define $\mathbf{P}$ on $\mathscr{F}$ by $\mathbf{P}(\mathrm{A}) = c \int\limits_\mathrm{A} f,$ with $c$ as stated in the theorem. Standard properties of Lebesgue integral show that $\mathbf{P}$ is indeed a probability measure (it is a positive measure with total mass equal to unity). To construct $X$ simply consider $X$ to be the identity function from $\Omega$ into $\mathbf{R}^d.$ Q.E.D.
Comment: There is very little intuition behind this construction except that is a theorical results that says the classical probability/statistics taught in first year college (all by densities) actually is a particular case of the abstract probability theory of graduate school using measure theory as backbone; so what is taught and done is very well justified by measure theory which is really quite flexible. What I think is really quite nice of the explicit construction is that it shows that the concept of random vector (which is primordial in elementary courses) actually is somewhat arbitrary (literally we are using the identity function as random vector). What really is important is that a concept such as a normalised positive measure captures very well the intuitive meaning of chances with the frequency intuition ("we say something has $x\%$ chance of occuring if in the long run of repeated experiments, about $x\%$ of them are success"). Furthermore, it also shows that the same space of measurable sets, namely the pair $(\Omega, \mathscr{F})$, is really meaningless probabilistically until we add a nice probability measure on top of it.
