Low-cost sensors for simultaneous localisation and mapping (SLAM) on robotic platforms (e.g. miniature sonar or radar) are susceptible to false and missed detections. This paper presents an occupancy-grid algorithm for SLAM which deals with this type of imperfect sensor measurements using the random finite set theoretical framework. The solution is formulated as a Rao-Blackwellised particle filter, where the robot pose is estimated using the sequential Monte Carlo method, while the map (occupancy-grid) update is calculated analytically. The particle filter is implemented using an adaptive importance sampling scheme with progressive correction. Results obtained in numerical simulations demonstrate a robust performance in the presence of false detections and low probability of detection.