Let
- $\lambda$ denote the Lebesgue measure on $\mathcal B(\mathbb R)$
- $p:[0,1)\to[0,\infty)$ be Borel measurable with $$\int_{[0,\:1)}p\:{\rm d}\lambda=1\tag1$$ and $$\mu:=p\left.\lambda\right|_{[0,\:1)};$$
- $(\Omega,\mathcal A,\operatorname P)$ be a probability space;
- $X$ be an $[0,1)$-valued random variable on $(\Omega,\mathcal A,\operatorname P)$ with $$X\sim\mu;$$
- $k\in\mathbb N$.
I would like to dividide $[0,1)$ into $k$ bins and project $X$ onto the left ende of its corresponding bin. This projection should be given by $$Y:=\frac{\lfloor kX\rfloor}k.$$ Does $Y$ still admit a density with respect to $\left.\lambda\right|_{[0,\:1)}$?
Consider $Z:=\lfloor kX\rfloor$. Since $$\lfloor kx\rfloor=i\Leftrightarrow x\in\left[\frac ik,\frac{i+1}k\right)\tag2$$ for all $x\ge0$, we should have $$\operatorname P\left[Z=i\right]=\int_{\left[\frac ik,\:\frac{i+1}k\right)}p\:{\rm d}\lambda\tag3$$ for all $i\in\{1,\ldots,k-1\}$. While $Z$ is $\{1,\ldots,k-1\}$-valued, we can clearly consider it as a real-valued random variable. But does it (and hence $Y$) admit a density with respect to $\left.\lambda\right|_{[0,\:1)}$?
Context: I'm asking this question cause I would like to use the Metropolis-Hastings algorithm on $[0,1)$, but project the proposals onto bins given as described in the question. Since the Metropolis-Hastings algorithm requires that the proposal kernel has a density with respect to the measure with respect to which the target density is given, I need to know whether a density exists or not. (If not, I would be thankful for notes on suitable variants of the Metropolis-Hastings algorithm.)
No, $Y$ does not admit a density with respect to Lebesgue measure. The support of $Y$ is $\{\frac{0}k,\frac1k,\dots,\frac kk\}$, which is a Lebesgue null set. If $Y$ had a density, then $\int_{S} f_Y(y)\,dy$ would be $1$, where $S$ is the support of $Y$, but since $\lambda(S)=0$, this is impossible.