On a property of joint terminal distributions of locally integrable increasing processes and their compensators

In this paper we prove that a joint distribution of a locally integrable increasing process $X^{\circ}$ and its compensator $A^{\circ}$ at a terminal moment of time can be realized as a joint terminal distribution of another locally integrable increasing process $X^{\star}$ and its compensator $A^{\star}$, $A^{\star}$ being continuous.