Let $X$ be a measurable space and $P_1, \dotsc, P_K$ be probability measures on it. If $\mu$ is a reference $\sigma$-finite measure with respect to which all $P_k$ are absolutely continuous, this machine learning paper defines strict linear independence by the property that for every $\lambda \in \mathbb R^K\setminus \{0\}$ it holds that $$ \int\left| \sum_{k=1}^K \lambda_k \frac{\mathrm dP_k}{\mathrm d\mu} \right| \mathrm{d}\mu \neq 0, $$ which can be rewritten as $\left|\sum_k\lambda_k P_k\right|(X) \neq 0$ using standard rules of Radon-Nikodym differentiation.
Writing $Q = \sum_{k=1}^K \lambda_k P_k$ and using this result, this should be equivalent to the following:
Distributions $P_1, \dotsc, P_K$ are strictly linearly independent iff for each $\lambda\neq 0$ there exists a measurable set $A$ such that $\lambda_1 P_1(A) + \cdots + \lambda_K P_K(A) \neq 0.$
This looks like a pretty general definition and I'm sure it must have been defined and investigated by measure theorists or information geometers. At the same time, I can't find any reference on this topic. I wonder if you could recommend books or academic publications investigating this definition and its implications.
