We are interested in Filippov systems which preserve a probability measure on a compact manifold. We define a measure to be invariant for a Filippov system as the natural analogous definition of invariant measure for flows. Our main result concerns Filippov systems which preserve a probability measure equivalent to the volume measure. As a consequence, the volume preserving Filippov systems are the refractive piecewise volume preserving ones. We conjecture that if a Filippov system admits an invariant probability measure, this measure does not see the trajectories where there is a break of uniqueness. We prove this conjecture for Lipschitz differential inclusions. Then, in light of our previous results, we analyze the existence of invariant measures for many examples of Filippov systems defined on compact manifolds.

我们对Filippov系统感兴趣，该系统在紧凑流形上保留了概率测度。我们将Filippov系统不变的度量定义为流量不变度量的自然相似定义。我们的主要结果与Filippov系统有关，该系统保留了与体积度量相当的概率度量。结果，体积保留的Filippov系统是折射的分段体积保留的系统。我们推测，如果Filippov系统接受不变概率测度，则该测度不会看到存在唯一性中断的轨迹。我们证明了Lipschitz微分包含的猜想。然后，根据我们之前的结果，我们分析了在紧凑流形上定义的Filippov系统的许多示例的不变测度的存在。