When a strict local martingale is projected onto a subfiltration to which it is not adapted, the local martingale property may be lost, and the finite variation part of the projection may have singular paths. This phenomenon has consequences for arbitrage theory in mathematical finance. In this paper it is shown that the loss of the local martingale property is related to a measure extension problem for the associated Föllmer measure. When a solution exists, the finite variation part of the projection can be interpreted as the compensator, under the extended measure, of the explosion time of the original local martingale. In a topological setting, this leads to intuitive conditions under which its paths are singular. The measure extension problem is then solved in a Brownian framework, allowing an explicit treatment of several interesting examples.