= Small-ball conditioning for a Poisson random measure
{title2=$\lim_{r\downarrow0}\mathbb E[e^{-M(f)}\mid M(B(0,r))\geq1]=e^{-f(0)}\mathbb E[e^{-M(f)}]$}
For a <Poisson random measure> on $\mathbb R^d$ with <Lebesgue measure> intensity and a nonnegative compactly supported <continuous function> $f$, put $m_r=\mu(B(0,r))$ and $a_r=\int_{B(0,r)}e^{-f}d\mu$. Continuity gives $a_r/m_r\to e^{-f(0)}$. With $L=\mathbb E e^{-M(f)}$, the <count-weighted Laplace functional of a Poisson random measure> and independent inside/outside counts give respectively
$$
\frac{\mathbb E[M(B(0,r))e^{-M(f)}]}{\mathbb P(M(B(0,r))\geq1)}=L\frac{a_r}{1-e^{-m_r}},\qquad
\mathbb E[e^{-M(f)}\mid M(B(0,r))\geq1]=L\frac{1-e^{-a_r}}{1-e^{-m_r}}.
$$
Both converge to the displayed value. The conditioning is meaningful for every $r>0$; it is not conditioning on a positive-probability point at the origin.
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