# 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 $$[x]=[y]$$.

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