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$\newcommand\R{\mathbb{R}}$We know that if $X\subseteq\R^n$ is a measure space, then the measure of any open ball $B_\varepsilon(x)\subset X$ is nonzero.

Is this true for a general measure space, one that is not necessarily defined on $\R^n$?

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First, there is no notion of "ball" in an arbitrary measure space.

Second, if $X$ is a metric space with a measure $\mu$, yes it is possible for a ball to have zero measure even in $\mathbb R^n$. For instance in $\mathbb R$ you could define $\mu(E) = \lambda(E \cap [0,1])$ where $\lambda$ is Lebesgue measure.

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