Let $\mu$ be a non-degenerate Gaussian measure on a seperable Hilbert space $\mathcal{H}$. Let $A \subset \mathcal{H}$ be a hyperplane. I would like to show that $\phi(A)$ has measure $0$ for a smooth map $\phi: \mathcal{H} \to \mathcal{H}$.
It is obvious that $A$ has measure $0$ since we can represent it as $$A = \{x \in \mathcal{H} | \langle l, x \rangle = c\}$$ for some $l \in \mathcal{H}, c \in \mathbb{R}$. Since $\langle l, \cdot \rangle$ is a non-degenerate Gaussian, the event $\{\langle l, x \rangle = c\}$ has measure $0$.
How would one prove that $\phi(A)$ has measure 0 too?