Superconducting qubits have emerged as leading candidates as the foundation of quantum information processing systems. Progress in superconducting qubit experiments with greater numbers of qubits and advanced techniques such as feedback will require faster and more accurate quantum state measurement. In particular, cyclic fault tolerance protocols such as the surface code require high accuracy measurement on time scales significantly shorter than the coherence times of the qubits. We have designed a multiplexed measurement system with a bandpass filter that allows fast measurement without increasing environmental damping of the qubits. We use this to demonstrate simultaneous measurement of four qubits on a single superconducting integrated circuit, finding that we can measured a single qubit state to 99.8% accuracy in 140 ns. This accuracy and speed is suitable for advanced multiqubit experiments including surface-code error correction.