This is a general question: If I want to show that my measure has a particular property, how can I do this? For instance, my measure is invariant under orthogonal transformations of my sets, what are my options to show this? Somehow I feel that I don't need to show this for all sets. At least, it should be sufficient to show this for a generating set of my $\sigma $algebra that is invariant under intersections. But I don't see, which theorems tells me that this is okay? Probably, it also makes a difference whether I am investigating $\sigma$- finite measures or general measures.
2026-04-06 01:50:18.1775440218
Show that measure has a particular property
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When the measure is $\sigma$-finite, $E=\cup E_n$, two measures $u,v$ on $E$ are the same iff $u$ coincides with $v$ on $E_n$ and on a $\pi$-system generating the $\sigma$-algebra on $E$.