Cramer wold device

Write ${\mathrm{\Phi }}_{\rho }(\theta )=\int e^{i\theta \cdot x}\rho (dx)$. The map $p_{\theta }$ is continuous: $\mathrm{\mid }{p}_{\theta }(x)-p_{\theta }(y)\mathrm{\mid }\le ({\sum }_{j=1}^d\mathrm{\mid }{\theta }_j\mathrm{\mid })\mathrm{\parallel }x-y{\mathrm{\parallel }}_2$. Hence its pushforward is a Borel probability, and ${\mathrm{\Phi }}_{\rho }(\theta )={\phi }_{(p_{\theta }{)}_{\ast }\rho }(1)$. In particular $\widehat{\rho }(\xi )={\mathrm{\Phi }}_{\rho }(-2\pi \xi )$. Positive probability measures have total variation one, since the absolute masses of any measurable partition sum to one. Thus if all projection laws of $\rho$ and $\tau$ agree, their finite-dimensional Fourier transforms agree at every $\xi$; the finite-measure uniqueness theorem gives $\rho =\tau$. The zero projection has the law of the constant zero and introduces no exception.

Euclidean space is complete. The set ${\mathbb{Q}}^d$ is countable by induction using the product theorem, and dense: approximate each of the finitely many coordinates of $x$ within $\eta \mathrm{/}(2d)$ by a rational to get a vector within Euclidean distance $\eta$. Thus ${\mathbb{R}}^d$, and in particular $\mathbb{R}$, is Polish. For each coordinate vector $e_j$, the assumed convergence and the tightness corollary give a compact real set with uniform complement mass below $\epsilon \mathrm{/}(2d)$ for all projected laws. Enlarge each such bounded compact set to $[-R_j,R_j]$. For the compact box $K={\prod }_{j=1}^d[-R_j,R_j]$, finite subadditivity yields ${\mu }_n(K^c)\le {\sum }_{j=1}^d{\mu }_n\{\mathrm{\mid }{x}_j\mathrm{\mid }>R_j\}<\epsilon \mathrm{/}2<\epsilon \mathrm{.}$ This proves tightness of the original laws, not just of their projections.

Prokhorov gives a weakly convergent further subsequence from every subsequence; write one such limit as $\nu$. For fixed $\theta$, continuous mapping gives convergence of its projected laws to $(p_{\theta }{)}_{\ast }\nu$, whereas the hypothesis gives convergence to $(p_{\theta }{)}_{\ast }\mu$. Apply the one-dimensional forward theorem to these two convergences at frequency one: the same numerical sequence has limits ${\mathrm{\Phi }}_{\nu }(\theta )$ and ${\mathrm{\Phi }}_{\mu }(\theta )$, so they are equal. This is true for every $\theta$. The Fourier identity and uniqueness argument of step 1.1 give $\nu =\mu$.

If convergence failed for a bounded continuous real test $f$, some positive error threshold would be exceeded at infinitely many indices. List those indices increasingly using the least next one. Step 2.1 supplies a further weakly convergent subsequence with limit $\mu$, contradicting that fixed error bound. Hence all such tests converge, which is ${\mu }_n\Rightarrow \mu$. AC is inherited from tightness/Prokhorov and finite-dimensional Fourier uniqueness, including its countable-choice transform and smoothing prerequisites; only finitely many coordinate choices are made locally. For $d=1$ the argument is unchanged. For $d=0$ the space is one point with zero metric and its only probability is unit mass there, so equality and convergence are immediate without a maximum over an empty coordinate set. Point masses in positive dimension are also covered.