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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:

CDF

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?

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  • $\begingroup$ You can only ask about existence of RND w.r.t. a given reference measure. Every measure is absolutely continuous w.r.t. itself with density $1$ so your question does not make sense.Also do not use the same symbols $X$ for a random variable and a real variable used to define its distribution function. $\endgroup$ – Kavi Rama Murthy Nov 25 '18 at 5:14
  • $\begingroup$ Thanks for the reply, what if the reference measure was the sum of lebesgue measure and a dirac measure on the set $\{0.8\}$? $\endgroup$ – InfProbX Nov 25 '18 at 10:40
  • $\begingroup$ If that is the reference measure then we do have absolute continuity and hence RND exists. $\endgroup$ – Kavi Rama Murthy Nov 25 '18 at 11:34
  • $\begingroup$ Could you please post a proof or a very rough sketch and what the RND looks like? I'd be really grateful!! $\endgroup$ – InfProbX Nov 25 '18 at 11:35
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    $\begingroup$ The RND is $0$ for $x<0$ as well as $x>0.8$, $2x$ for $0\leq x <0.8$ and $[1-(0,8)^{2}]$ for $x=0.8$ $\endgroup$ – Kavi Rama Murthy Nov 25 '18 at 11:43

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