If $x,y \in X$, then $[x] \cap [y] =\emptyset$ or $[x]=[y]$ for $ [x]= \cap \lbrace A \in \mathcal{M} \:|\: x \in A \rbrace$.

by Cos   Last Updated September 11, 2019 18:20 PM - source

Let $X$ be a non empty set, and $(X, \mathcal{M})$ a measurable space. Then I define

$$ [x]= \cap \lbrace A \in \mathcal{M} \:|\: x \in A \rbrace$$.

At my class my teacher told us to prove that:

If $x,y \in X$, then $[x] \cap [y] =\emptyset$ or $[x]=[y]$.

I have just made a simple attempt to this as if $[x] \cap [y] =\emptyset$ we are done if $[x] \cap [y] \neq \emptyset$ but Im having issues to continue here as Im not sure how to express $u \in [x] \cap [y]$ in order to conclude $[x]=[y]$.



Related Questions


Implications with the Lebesgue integral

Updated July 21, 2018 00:20 AM

Does this kind of $C^1$-diffeomorphism a contraction?

Updated December 16, 2017 04:20 AM

Lebesgue integration of simple functions

Updated April 22, 2015 22:08 PM

Application of Monotone Convergence Theorem

Updated April 29, 2015 01:08 AM