I have a basic question about Gaussian measures on a Hilbert space:

Let $\mu$ be a non-degenerate Gaussian measure on a Hilbert space $(H_0,\left\langle \cdot,\cdot \right\rangle_0)$. Then the covariance operator $S$ of $\mu$ is a bijective, non-negative, self-adjoint trace-class operator on $H_0$. Therefore, $$ \left\langle x,y \right\rangle_1 := \left\langle \sqrt{S} x, \sqrt{S} y \right\rangle_0, \qquad x,y \in H_0 $$ defines another inner product on $H_0$ which is weaker than $\left\langle \cdot,\cdot \right\rangle_0$, as for the induced norms we have that $\left\lVert \cdot \right\rVert_1 \leq \left\lVert T \right\rVert_{op} \left\lVert \cdot \right\rVert_0$. Let $H_1$ be the completion of $H_0$ with respect to $\left\lVert \cdot \right\rVert_1$. Then $H_0 \subseteq H_1$ densely and as a Borel set.

Let $\mathcal{B}(H_0)$ and $\mathcal{B}(H_1)$ denote the Borel sets of $H_0$ and $H_1$, with respect to the norms $\left\lVert \cdot \right\rVert_0$ and $\left\lVert \cdot \right\rVert_1$. Then $\mathcal{B}(H_1) \cap H_0 \subseteq \mathcal{B}(H_0)$, where $\mathcal{B}(H_1) \cap H_0 = \{ A \cap H_0 \colon A \in \mathcal{B}(H_1) \}$

**My question is:**
If a probability measure $\nu$ defined on $(H_0,\mathcal{B}(H_0))$ agrees with $\mu$ on $\mathcal{B}(H_1)\cap H_0$, does it follow that $\nu=\mu$, i.e. that those measures also agree on $\mathcal{B}(H_0)$?