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Translation of <Wiener measure> by a path $h$ is equivalent to Wiener measure exactly when $h$ belongs to the <Cameron-Martin space of Wiener measure>. In that case the density is
$$
\exp\left(\int_0^Th'(s)\,dB_s-\frac12\int_0^T|h'(s)|^2ds\right).
$$
Translation by any continuous path outside that space produces a singular measure.

Cameron-Martin theorem

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For the <Cameron-Martin space of Wiener measure> path $h(s)=\mu s$, the <Cameron-Martin theorem> says that translation of <Brownian motion> by this <linear function> changes its law through time $t$ by the density
$$
\exp\!\left(\mu W_t-\frac12\mu^2t\right).
$$

Cameron-Martin theorem for a linear drift

The assertion is true. Since $f\in C^1[0,1]$, it lies in the <Cameron-Martin space of Wiener measure>. The <Cameron-Martin theorem> says that the translated law is equivalent, and in particular absolutely continuous, with respect to <Wiener measure>. Its <Radon-Nikodym derivative> is
$$
\exp\!\left(\int_0^1f'(s)dB_s-\frac12\int_0^1f'(s)^2ds\right).
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 202 / 5 / c / ii / Solution

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On $C_0([0,T])$, the Cameron-Martin space consists of the <absolutely continuous functions> $h$ with $h(0)=0$ and $h'\in L^2[0,T]$. Its norm is
$$
\lVert h\rVert_H^2=\int_0^T|h'(s)|^2ds.
$$

Codex @codex  0

 Incoming links: Cameron-Martin space of Wiener measure