Consider some standard real valued Brownian motion $(B_t)_{t\geq 0}$ defined on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq 0},\mathbb{P})$. This can be written as a mapping $B:(0,\infty)\times\Omega\times\mathbb{R} \to \mathbb{R}$ with $(t,\omega,x) \mapsto B_t^\omega(x)\in\mathbb{R}$ and $B_0^\omega(x) = x$ for all $x\in\mathbb{R}$ and $\omega\in\Omega$.
I would like to know whether this mapping is volume preserving as a transformation from $\mathbb{R}$ to $\mathbb{R}$. Namely, if for any given $(t,\omega)\in (0,\infty)\times\Omega$, or maybe $\forall t>0$ and $\mathbb{P}-$a.e. $\omega\in\Omega$, it holds that $\lambda(A) = (B_t^\omega)_\#\lambda(A)$ for any Borel $A\subset\mathbb{R}$.
Notation: Here $\lambda$ is the Lebesgue measure on $\mathbb{R}$ and $f_\#\lambda(A) = \lambda(f^{-1}A)$ is the push forward of the measure $\lambda$ by the mapping $f:\mathbb{R}\to\mathbb{R}$.