Cameron--Martin formula for the $\sigma$-finite measure unifying Brownian penalisations

Quasi-invariance under translation is established for the $\sigma$-finite measure unifying Brownian penalisations, which has been introduced by Najnudel, Roynette and Yor. For this purpose, the theory of Wiener integrals for centered Bessel processes, due to Funaki, Hariya and Yor, plays a key role.