Support under a continuous map ($\mathrm{supp}(f_{\ast }\mu )=\overline{f(\mathrm{supp}\mu )}$)

For a Borel probability measure on a Polish space and a continuous map to another Polish space, the image measure has the displayed support. Every neighborhood of an image of a support point has an open inverse image of positive measure. Conversely, the closed image closure has full image measure because the original support has full measure. This transfers a support description for a driving rough path to the solution of a rough differential equation.