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Let $\mu :\mathcal{B}(\mathbb{R}) \to [0,\infty]$ be a measure such that for all $h\in \mathbb{R}$ and $A\in \mathcal{B}(\mathbb{R})$, $\mu(A+h) = \mu(A)$. Let $c= \mu((0,1])$.

I want to prove the property that $\mu((0,x]) = cx$ for any $x\geq 0$.

I have already proven that $\forall x\geq 0, \forall q\in\mathbb{N}: \mu((0,qx]) = q\mu((0,x])$.

As well as $\forall p,q\in \mathbb{N}: \mu((0,\frac{p}{q}]) = c\frac{p}{q}$.

However now I'm stuck. I know that $\mu$ is a measure and therefore $\sigma$-additive, however I am not quite sure how to use this because $x$ is now a real number instead of an integer or a fraction.

T C Molenaar
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  • $\mu$ is a measure, so the function $t\mapsto\mu((0,t])$ satisfies some very simple general properties that can help you here. Something as basic as being non-negative, but different... – kimchi lover Nov 26 '17 at 16:48
  • @ZacharySelk I am asking how to prove it using only the definition of measure and that $\mu$ is invariant, the question there is what the properties are and which invariant measures exist on $\mathbb{R}$. – T C Molenaar Nov 26 '17 at 16:52
  • @kimchilover I don't see what you mean exactly and how this helps me. – T C Molenaar Nov 26 '17 at 16:53
  • Read the top answer and it is a solution to your problem –  Nov 26 '17 at 16:54
  • @ZacharySelk I see, however it is not quite clear to me what is meant by 'use the monotonicity of the measure to get lower continuity of the measure for all intervals $[a,b)$'. I know what the monotonicity of the measure entails, however I am not sure what is meant by 'a lower continuity' and how I can conclude from this that it is true for all intervals $(a,b]$. – T C Molenaar Nov 26 '17 at 17:02

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