This paper introduces a new Kalman filter-based receiver autonomous integrity monitoring (RAIM) method for navigation systems that require measurement filtering over time (such as integrated GPS/INS navigation or carrier phase positioning). In contrast with existing sequential fault-detection algorithms, the proposed method is computationally efficient, straightforward to implement, and most importantly, it enables direct integrity risk evaluation. First, the Kalman filter-based detection test-statistic is established as a sum of generalized non-central chi-square distributed random variables. This test statistic can be recursively updated in real-time by simply adding the current-time Kalman filter residual contribution to a previously computed weighted norm of past-time residuals. Second, the test statistic is proved to be stochastically independent from the state estimate error, which enables to rigorously quantify the integrity risk (the integrity risk is defined as the probability of faults going undetected while causing hazardous information). The Kalman filter-based RAIM method is presented in a general formulation applicable to linear dynamic systems. The performance of the new detection method is illustrated and analyzed in a benchmark application of aircraft precision approach, where differential GPS and Galileo code and carrier phase measurements are filtered for positioning and floating cycle ambiguity estimation.