Timeline for Separable sigma-algebra: equivalence of two definitions
Current License: CC BY-SA 2.5
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| Jun 8, 2010 at 21:29 | comment | added | coudy | @Kestutis. I should have said: the first definition is useful when studying Borel sets, but is too restrictive when studying e.g. Lebesgue measurable sets. Now for the equivalence, this comes from the fact that any subset (in our case {1_A, A measurable}) of a separable metric space (in our case L^1) is separable. And from the fact that 1_{A_n} converges to 1_A in L1 norm iff A=limsup A_n=liminf A_n up to a negligible set. See e.g. the book of Dudley, "Real analysis and probability". | |
| Jun 8, 2010 at 20:59 | comment | added | Kestutis Cesnavicius | Why does the first definition only makes sense for $\sigma$-algebra of Borel sets? Why is the second definition \emph{equivalent} to saying that the algebra is generated by a countable collection of subsets, together with the null sets? | |
| Jun 8, 2010 at 19:25 | history | answered | coudy | CC BY-SA 2.5 |