I just started self-studying measure theory and had a question about the "generalised density" (as it appears on Wikipedia):
A random variable X with values in a measurable space $({\mathcal{X}},{\mathcal {A}})$ (usually $\mathbb {R} ^{n}$ with the Borel sets as measurable subsets) has as probability distribution the measure $X_*P$ on $({\mathcal {X}},{\mathcal {A}})$: the density of $X$ with respect to a reference measure $\mu$ on $({\mathcal {X}},{\mathcal{A}})$ is the Radon–Nikodym derivative:
$$f={\frac {dX_{*}P}{d\mu }}.$$
For a random variable $X$, if we had a cumulative distribution function:
$$F(X) = \begin{cases} 0 & if \; X < 0 \\ X^2 & if \; 0 \leq X < 0.8 \\ 1 & if \; X \geq 0.8 \end{cases}$$
... that looks like:
Then, does a "generalised density" or a Radon-Nikodym derivative as defined above exist?
I think that it does... If $\mathcal X = [0, 0.8) \cup \{0.8\}$, for an appropriate $\sigma$-algebra $\mathcal A$, could one use the "reference measure":
$$\mu = l + \delta_{0.8}$$
... where $l$ is the Lebesgue measure and $\delta_{0.8}$ is the Dirac measure on $\{0.8\}$? I'd also love some help clarifying what $X_*P$ is (is this a "pushforward measure", and if so, is this the same as the "measure induced by the CDF"?)
If one can use the reference measure above, why can we use it? And would the Radon-Nikodym derivative be:
$$f = \frac{dX_*P}{d\mu} = 2X * \mathcal I(X \in [0, 0.8)) + 0.2 * \mathcal I(X = 0.8)$$
where $\mathcal I$ is the indicator function?