A new kind of augmentation of filtrations suitable for a change of probability measure by a strict local martingale

In this note we introduce a new kind of augmentation of filtrations along a sequence of stopping times. This augmentation is suitable for the construction of new probability measures associated to a positive strict local martingale as done in Kardaras et al. (2015), while it is on the other hand rich enough to make classical results from stochastic analysis hold true on some stochastic interval of interest.