Pointwise information entropy for metric spaces

We give a definition of information entropy for points in metric spaces. This entropy measures the amount of information needed to specify a point of a metric space. For the definition of entropy we need to introduce an additional structure on the metric space which will be called a computable structure. If information sources are considered as a metric space we prove that our entropy has the same value as Shannon's one for almost all points of the space. If general metric spaces are considered there is a relation between entropy and dimension.