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If $X$ is a metric space and $0<\mu(X)$, where $\mu$ is a radon measure and $\mu(\{x\})=0$. Can we always split $X = X_1 \sqcup X_2$ into two disjoint sets where $\mu(X_1) = a$, for any $0<a< \mu(X)$?

Or more generally does there always exist a set $X_1$ such that $\mu(X_1) = a$

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No, for this we need a criterion for the measure, called sometimes "atomless". –  GEdgar Nov 28 '12 at 0:56
    
What about if we let $\mu({x}) = 0$ for every $x\in X$ –  sadsd Nov 28 '12 at 0:58
    
In a metric space (where every singleton is a measurable set), $\mu(\{x\}) = 0$ for every $x$ is known as "atomless". Now in a really big metric space, you could have only measures $0$ and $\infty$, but I believe that is not allowed when you say "Radon measure". –  GEdgar Nov 28 '12 at 15:30
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1 Answer

First note that your measure has the following property, a consequence of the fact that Radon measures are locally finite:

($*$) If $x\in X$, then $\lim_{r\to 0^+}\mu(B(x,r)) = 0$.

With this, you can prove what you want with a Zorn's lemma argument (sorry, there is almost certainly a better way), which goes as follows. First, let $T$ be the collection of all open subsets $U$ of $X$ with measure $\mu(U)\leq a$. Partial order $T$ by inclusion. First we must show $T$ is nonempty. If $x$ lies in the support of $\mu$, then every ball around $x$ has positive measure (by definition of the support of $\mu$), and property ($*$) tells us that there are balls around $x$ of as small (positive) measure as we like. Thus $T$ is nonempty. If $\{U_\alpha\}$ is a totally ordered subset of $T$, then $\bigcup U_\alpha$ is an upper bound of $\{U_\alpha\}$ in $T$. Thus we can apply Zorn to get a maximal element $U$ of $T$. I claim that $\mu(U) = a$. If $\mu(U)<a$, then, since $\mu(U)<\mu(X)$, there is a point $x\notin U$ that lies in the support of $\mu$. By choosing $r>0$ small enough, $\mu(U) + \mu(B(x,r))<a$ by property ($*$). But then $U\cup B(x,r)\in T$ strictly contains $U$, contradicting the maximality of $U$.

This argument proves that you can find an open set $U$ with $\mu(U) = a$, and that in fact you can choose $U$ to be a maximal open set with this property.

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Just for clarity, $\mu$ nonatomic is needed for (*). And you need an extra argument why the support of $\mu$ is not empty (otherwise every compact set has zero measure, so by inner regularity $\mu$ is the zero measure). –  Thomas Nov 28 '12 at 9:09
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