Wiener integral for the coordinate process under the $ σ$-finite measure unifying Brownian penalisations

Wiener integral for the coordinate process is defined under the $ \sigma $-finite measure unifying Brownian penalisations, which has been introduced by Najnudel, Roynette and Yor. Its decomposition before and after last exit time from 0 is studied. This study prepares for the author's recent study of Cameron--Martin formula for the $ \sigma $-finite measure.