We prove that averaging operators are uniformly bounded on $L^1$ for allgeometrically doubling metric measure spaces, with bounds independent of themeasure. From this result, the $L^1$ convergence of averages as $r o 0$immediately follows.
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机译：我们证明了平均运算符以$ l ^ 1 $统一地绑定，以便为成绩加倍的度量标准度量空间，带有无关的界限。从这个结果来看，$ l ^ 1 $ $ r r to 0 $紧立立即跟随。
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