As a follow up to [this question](https://mathoverflow.net/questions/199020/volume-of-randomly-changing-sphere-follows-beta-distribution), concerning [this paper](http://dl.acm.org/citation.cfm?id=355745):


Given random variables $X\_1,\ldots,X\_N,X\_q:\Omega\rightarrow\mathbb{R}^d$, where $X\_1,\ldots,X\_N$ are independent and identical distributed. Draw an $\omega\in\Omega$. We now want to pick the $k$ points from $X\_1(\omega),\ldots,X\_N(\omega)$ that are closest to $X\_q(\omega)$. Wlog assume these are $X\_1(\omega),\ldots,X\_k(\omega)$. Define
$$B\_k(X\_q(\omega))=\{x\in\mathbb{R}^d\mid D(x,X\_q(\omega)\leq D(X\_k(\omega),X\_q(\omega))\}$$
where $D(x,y)$ is some dissimilarity measure (think of a metric). The volume of this ball is
$$v\_k(X\_q(\omega))=\int\_{B\_k(X\_q(\omega))}1\:dx.$$
Let $\delta(x)$ be the probability density function of the $X\_i$, then let
$$u\_k(X\_q(\omega))=\int\_{B\_k(X\_q(\omega))}\delta(x)\:dx$$
be the probability content of the ball $B\_k(X\_q(\omega))$. Obviously we have $0\leq u\_k(X\_q(\omega))\leq 1$.


Now assume $N$ large enough such that $\delta(x)$ can be approximated by a constant $\overline{p}(X\_q(\omega))$ on $B\_k(X\_q(\omega))$.


Fianlly, the authors of the paper state that $\overline{p}(X\_q(\omega))\neq 0$, but I do not see why this should hold. I though that
$$\overline{p}(X\_q(\omega))\approx\int\_{B\_k(X\_q(\omega))}\delta(x)\:dx.$$
which can be $0$ for example if $\delta(x)$ is not continuous, but discrete.
Can someone explain, why $\overline{p}(X\_q(\omega))\neq 0$?



Probability Content of a random ball in R^n
As Joe Neeman pointed out in the comment on the OP:



> 
> The very fact that they're aksing to approximate a function $\delta(x)$ by a constant in a small ball already assumes that the function is continuous.
> 
> 
> 


We now approximate $\delta(x)$ on $B\_k(X\_q(\omega))$ by $\overline{p}(X\_q(\omega))=\int\_{B\_k(X\_q(\omega))}\delta(x)\:dx$. Then $\overline{p}(X\_q(\omega))\neq0$ since $v\_k(X\_q(\omega))\neq0$, $\delta(x)$ continuous, and $X\_1(\omega),\ldots,X\_k(\omega)\in B\_k(X\_q(\omega))$. That is, there is a small neighborhood $U\_i$ around $X\_i(\omega)$, $i=1,\ldots,k$, such that $\delta(x)>0$ for all $x\in U\_i$.


