Let us call a measure $\Lambda$ homogeneous if there is an $\epsilon>0$ so that for all $r>0$ and $x,y$ in the support of $\Lambda$, we have

$$\Lambda(B(x,r))>\epsilon\Lambda(B(y,r))$$

where as usual $B(x,r)$ is the ball of radius $r$ centred at $x$. I am thinking of $\mathbb{R}^d$ but it could be a general metric space. Q: Is this (equivalent to) a known definition in the literature?

Update: I followed up dirk's comment to user116082's answer. Google scholar gives me exactly 10 results for "quasi uniform measure." The two that seem relevant are

Saul Jacka and Jon Warren, Random orderings of the integers and card shuffling, Stochastic Processes and their Applications, 117, 708-719 (2007).

which is quite a different concept. Also

É Janvresse and T. de la Rue, A class of pairwise-independent joinings. Ergodic Theory and Dynamical Systems, 28, 1545-1557 (2008).

which is very close to what I have applied to cylinder sets in symbolic dynamics. The latter paper gives the definition but is worded in a way that suggests it is a known concept. The relevant reference [5] in that paper does not appear to contain this terminology, though (searching using google books).

So, for now, the original question still stands, for more general contexts than symbolic dynamics, and also Q2: What is the first usage of the term quasi-uniform measure as defined in the second paper above?