Using Riesz’s lemma, we may construct a sequence such that and for . Then, the balls are all disjoint and contained in . If the measure is finite on open balls, it must follow, by translation-invariance and -additivity, that the measure of any ball of radius is zero, which implies that the measure is trivial.
The Noriega Book 
Using Riesz’s lemma, we may construct a sequence
such that 
and 
for 
. Then, the balls 
are all disjoint and contained in 
. If the measure is finite on open balls, it must follow, by translation-invariance and 
-additivity, that the measure of any ball of radius 
is zero, which implies that the measure is trivial.
LikeLiked by 1 person