Remark

— If S is the support of $\mu$, then $S \cap X$ (which is closed in X) contains the support of $\mu_X$ by Cor. 2, but may be distinct from it. For example, if $\mu$ is a diffuse measure and X is a subspace reduced to a point, then the induced measure $\mu_X$ is zero, hence its support is empty. Note, however, that the support of $\mu_X$ is equal to $S \cap X$ if X is open in T.