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In his book ‘Real Analysis’, Folland makes a comment concerning the definition of a Radon measure that I don’t quite understand. He defines a Radon measure as a Borel measure which is finite on compact sets, outer regular on every Borel set and inner regular on every open set. After the definition, he says that “It turns out that regularity is a bit too much to ask for when $X$ is not $\sigma-$compact.”

Can someone explain this statement? The only thing I know is that In a later result it is shown that in a $\sigma-$compact space every Radon measure is regular.

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I don't know whether this is an explanation oof that statement or not, but here's an example of a non-$\sigma$-finite Radon measure that's not regular:

Define $X$ and $\mu$ as here. Then $\mu$ is not inner regular.

Let $E=\{0\}\times[0,1]$. Then $\mu(E)=\infty$, but if $K\subset E$ is compact then $K$ is finite, hence $\mu(K)=0$.

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