Consider an autonomous system described by \begin{equation} \dot{x}=F(x), \end{equation} where $F\in C^1(X)$. Define the semigroup of transformations $\{\mathcal{S}_t\}_{t\geq0}$ as $$\mathcal{S}_t(x_0)=x(t),$$ where $x(t)$ is the solution of the ODE above, corresponding to the initial condition $x(0)=x_0\in X$.
Definition. Given a measure space $(X,\mathcal{A},\mu)$, we say that a semigroup of transformations $\{\mathcal{S}_t\}_{t\geq0}$ is nonsingular if for all $t\geq0$ $$\mu(\mathcal{S}^{-1}_t(A))=0 \qquad \text{for each $A\in\mathcal{A}$ such that $\mu(A)=0$}.$$
(A semigroup of transformations is nonsingular as long as it does not map sets of measure non-zero to sets of measure zero.)
Conjecture. Let $X=\mathbb{R}^n$, $\mu$ be the Lebesgue measure, and $\mathcal{A}$ be the usual $\sigma$-algebra on $\mathbb{R}^n$. Then $\{\mathcal{S}_t\}_{t\geq0}$ is nonsingular if $F\in C^1(X)$.
How can I prove/disprove the above?