Definition of homogeneous or quasi-uniform or almost uniform measure

Question

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?

Update 2: Repeating the above process for "almost uniform measure" yields 11 google scholar results, one of which gives a definition that looks equivalent to the above, except for all points in the space, not just in the support of the measure.

For example, the usual measure on the middle third Cantor set satisfies the above definition in $\mathbb{R}$ but the "almost uniform measure" definition only if the space is the Cantor set itself with metric induced from its embedding into $\mathbb{R}$.

The relevant reference is

M. Studený, "On the differentiation theorem in metric groups." Commentationes Mathematicae Universitatis Carolinae 24 223-232 (1983).

On balance, it seems best to stick with "almost uniform measure" but note the above distinction with regard to the support. I am still interested in any further references or information about these definitions.

Answer 1

I can't comment, being anonymous, and this is not a true answer to the question.

Your notion is similar to the notion of a uniformly distributed measure used in geometric measure theory. That is a measure that assigns the (exact) same mass to all balls of the same radius.

Your measure only assigns uniformly comparable mass to all balls of the same radius. I might therefore call it ``quasi-uniformly distributed'' although this is a bit of a mouthful.