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Let $\mathcal A$ be a Borel $\sigma$-algebra on $[0, 1]$, and let's introduce a metric on it by $$ d(A, B) = \lambda(A\mathbin\Delta B) \qquad \forall A,B\in \mathcal A $$ where $\lambda$ is the Lebesgue measure and $\Delta$ is the symmetric difference. If we take a quotient space modulo $A\sim B \iff d(A,B) = 0$, do we get some other well-known metric space? Of course, it is a subspace of $L_1([0,1])$ , but I hoped for a more concrete description. I also think, that space may be somewhat smaller. For example, if we consider a discrete measure with support of $n$ points, then we get $\Bbb R^n$ for functions and $2^n$ for measures.

Michael Hardy
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SBF
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  • I changed $A\Delta B$ to $A\mathbin\Delta B$, coded as A\mathbin\Delta B. The visual difference is that the spaces to the left and right of $\Delta$ are the same as those with a plus or minus sign used as a binary operator, but \mathbin\Delta B with no preceding A appears as $\mathbin\Delta B$, just as $+5$ with nothing before the plus sign lacks the same spacing between the plus sign and the 5 that is seen in $3+5$. That's what \mathbin is for. $\qquad$ – Michael Hardy Jan 25 '16 at 17:17

1 Answers1

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I think that almost everything you might want to know about this space ($\mathrm{MALG}_\lambda$, called the Lebesgue measure algebra) is in exercises (17.42)-(17.46) in Kechris' book.

This metric $d$ turns $\mathrm{MALG}_\lambda$ into a Polish space, and the Boolean operations ($\cup$, $\cap$, and relative complement in $[0,1]$) are well defined and continuous.

Moreover, $\mathrm{MALG}_\lambda$ is the unique, up to iso, measure algebra which, as a $\sigma$-Boolean algebra

  1. is generated by a countable set and
  2. has no atoms.

As an aside, an analogous algebra can be defined in terms of category, but there is no Polish topology making the Boolean operations continuous for this one.

Pedro Sánchez Terraf
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