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A simple locally finite point measure on $[0,\infty )$ with void probability $e^{-\lambda (A)}$ on finite unions of bounded intervals is a Poisson random measure with Lebesgue measure intensity. Void-event intersections factor, hence the occupation indicators on disjoint cells are independent. The dyadic occupancy approximation of a simple point measure is binomial with rare-event probability $1-e^{-2^{-n}}$ and converges to a Poisson count of mean the interval length. Independence survives joint convergence for disjoint intervals. Extension of the mean measure and the Laplace functional of a Poisson random measure to Borel functions identifies all Borel counts, not just intervals.

Refining an occupied cell replaces it by two child cells, at least one of which is occupied. All children of a cell contained in $J$ remain contained in $J$, and additional cells can enter the sum. Hence $M_n(J)$ is nondecreasing, pathwise, and is at most $M(J)$.

We must also address boundary points for a fixed interval $J$. For every fixed deterministic $x$, take bounded intervals $I_r$ containing $x$ with lengths tending to zero. The assumed void law givesThus the fixed endpoints of $J$ are almost surely not atoms. This is the exponential void probabilities forbid fixed atoms observation; it is not a simultaneous assertion for all possible real endpoints.

In a fixed bounded interval containing $J$, local finiteness gives only finitely many atoms. Simplicity makes their locations distinct, so there is a positive minimum distance between any two, unless there is at most one. Every atom in the interior of $J$ also has positive distance from its endpoints. For sufficiently fine dyadic cells each such atom lies alone in a cell fully contained in $J$, including if the atom itself is at a dyadic boundary because the cells are half-open. Thus the dyadic occupancy approximation of a simple point measure eventually equals $M(J)$, almost surely. In particular,

The union of fully contained level-$n$ cells misses at most two boundary portions of total length at most $2^{1-n}$. Consequently $N_n{2}^{-n}\to \lambda (J)$, and $p_n/2^{-n}\to 1$. For $0\le z\le 1$, the probability generating function from the binomial calculation givesThe convergence follows by expanding the logarithm, since $N_n{p}_n\to \lambda (J)$ and $N_n{p}_n^2\to 0$. Since the counts converge almost surely, bounded convergence identifies the same expression as $\mathbb{E}[z^{M(J)}]$. It is the generating function of the Poisson distribution, whose power-series coefficients determine its law. Thus