Suppose $W_{n}(x)$ is a function from $\mathbb{R}^{n}$ to $\mathbb{C}$ which is positive-definite. By Bochner's Theorem, there exists an unique borel probability measure $\mu_{n}$ on $\mathbb{R}^{n}$ such that $$W_{n}(x) = \int_{\mathbb{R}^{n}}e^{i\langle y,x\rangle}d\mu_{n}(y)$$ Now, if $m > n$, we can apply the same reasoning. Besides, because of of the uniqueness of $\mu_{n}$ in Bochner's Theorem, if $x = (x_{1},...,x_{n},0,...,0)\in \mathbb{R}^{m}$ we would expect $$W_{m}(x) = \int_{\mathbb{R}^{m}}e^{i\langle y,x\rangle}d\mu_{m}(x) = W_{n}(x) \hspace{1cm} (1)$$ once $W_{m}$ does not have dependence in the $n+1,...,m$ entries of $x$.
$\textbf{Questions}$: 1) Does (1) really holds and how can I prove it? 2) If (1) holds, does this imply the sequence of measures $\{\mu_{n}\}_{n\in \mathbb{N}}$ is consistent?
REMARK: For completeness, a family $\{\mu_{n}\}_{n\in \mathbb{N}}$ of measures on $\mathbb{R}^{n}$ if $\mu_{m}(A\times\mathbb{R}^{m-n})=\mu_{n}(A)$ if $m>n$ and $A$ a borel set of $\mathbb{R}^{n}$.