Timeline for lebesgue measure and countable sets

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Mar 14, 2010 at 20:24	comment	added	Arturo Magidin		@curious: If all points had the same positive measure, then you would be correct: a countably infinite set would have infinite measure. However, there are measures in which (some) single points have positive measure, but countable sets may still have finite measure. For example, take a bijection between $f\colon\mathbb{Q}\to\mathbb{N}$, and define a measure $\mu$ on $\mathcal{P}(\mathbb{R})$ by letting $\mu(X) = \sum_{q\in \mathbb{Q}\cap A}2^{-f(q)}$. Then every subset of $\mathbb{R}$ is measurable, and has finite measure; some points (the rationals) have positive measure, some have measure 0.
Mar 14, 2010 at 0:06	comment	added	Gerald Edgar		Positive isn't enough, perhaps the measures of the points in some countable set are the terms of a convergent series.
Mar 13, 2010 at 21:44	comment	added	curious		so if the measure of a single point was not zero then measure of a countable set would be $\infty $ which would be the same as the measure of set $[0, \infty) which wouldnt make sense.
Mar 13, 2010 at 21:37	history	answered	Arturo Magidin	CC BY-SA 2.5