In mathematics, strict positivity is a concept in measure theory. Intuitively, a strictly positive measure is one that is "nowhere zero", or that is zero "only on points".

Let ${\displaystyle (X,T)}$ be a Hausdorff topological space and let ${\displaystyle \Sigma }$ be a ${\displaystyle \sigma }$-algebra on ${\displaystyle X}$ that contains the topology ${\displaystyle T}$ (so that every open set is a measurable set, and ${\displaystyle \Sigma }$ is at least as fine as the Borel ${\displaystyle \sigma }$-algebra on ${\displaystyle X}$). Then a measure ${\displaystyle \mu }$ on ${\displaystyle (X,\Sigma )}$ is called strictly positive if every non-empty open subset of ${\displaystyle X}$ has strictly positive measure.

More concisely, ${\displaystyle \mu }$ is strictly positive if and only if for all ${\displaystyle U\in T}$ such that ${\displaystyle U\neq \varnothing ,\mu (U)>0.}$

Van Casteren, J.A. (1994), Strictly Positive Radon Measures. Journal of the London Mathematical Society, 49: 109-123. doi: 10.1112/jlms/49.1.109