I am confused by the role Girsanov Theorem plays in deducing absolute continuity of laws of certain processes. Say $B$ is a Brownian motion in $(\Omega, \mathcal{F}_{\infty},\mathcal{F}_t,P)$ and let $K$ be a well-behaved enough process that $H_t=\int_0^t K_s ds$ and $N_t=\int_0^t K_s dB_s $ are both defined and well-behaved and $X_t=\exp(N_t -\frac{\langle N \rangle}{2})$ is UI. Then by applying Girsanov we find $Q$ such that $B'_t=(B_t - H_t)_{t \geq 0}$ is a local martingale under this new probability measure $Q$ - actually a Brownian motion. But then the following statements confuse me: apparently, the law of $(B_t-H_t)_{t\geq 0}$ under $P$ is absolutely continuous wrt the law of a Q-Brownian motion and, conversely, the law of $(B'_t+H_t)_{t \geq 0}$ under $Q$ is absolutely continuous wrt to the law of a $P$-Brownian motion.
From Girsanov theorem, the new probability measure $Q$ is absolutely continuous wrt $P$ by construction as far as I understood. Why then does this absolutely continuity of laws hold? Just because the underlying probability measures are absolutely continuous and the laws are just the push-forward of these probabilities?