= Support under a continuous map
{title2=$\operatorname{supp}(f_*\mu)=\overline{f(\operatorname{supp}\mu)}$}
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>.
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