Hello.
There are two statements below (assuming \(\displaystyle \operatorname{E}[X]\) and \(\displaystyle \operatorname{E}[Y]\) exists):
1. If \(\displaystyle X \le Y\) (a.s.) then \(\displaystyle \operatorname{E}[X] \le \operatorname{E}[Y]\).
2. If \(\displaystyle X = Y\) (a.s.) then \(\displaystyle \operatorname{E}[X] = \operatorname{E}[Y]\).
Can I remove the "a.s." condition and simply write "if \(\displaystyle X \le Y\) then \(\displaystyle \operatorname{E}[X] \le \operatorname{E}[Y]\)" and "If \(\displaystyle X = Y\) then \(\displaystyle \operatorname{E}[X] = \operatorname{E}[Y]\)"?
I think yes because part of the expected value that corresponds to subset \(\displaystyle A \subseteq \operatorname{image} X, \, P(A) = 0\) is equal to zero and doesn't change anything. Nevertheless, class of "almost surely" statements contains subclass of "ordinary" statements so the statements above are written with more general "a.s." condition.
Please correct me if I'm wrong. Thanks.
There are two statements below (assuming \(\displaystyle \operatorname{E}[X]\) and \(\displaystyle \operatorname{E}[Y]\) exists):
1. If \(\displaystyle X \le Y\) (a.s.) then \(\displaystyle \operatorname{E}[X] \le \operatorname{E}[Y]\).
2. If \(\displaystyle X = Y\) (a.s.) then \(\displaystyle \operatorname{E}[X] = \operatorname{E}[Y]\).
Can I remove the "a.s." condition and simply write "if \(\displaystyle X \le Y\) then \(\displaystyle \operatorname{E}[X] \le \operatorname{E}[Y]\)" and "If \(\displaystyle X = Y\) then \(\displaystyle \operatorname{E}[X] = \operatorname{E}[Y]\)"?
I think yes because part of the expected value that corresponds to subset \(\displaystyle A \subseteq \operatorname{image} X, \, P(A) = 0\) is equal to zero and doesn't change anything. Nevertheless, class of "almost surely" statements contains subclass of "ordinary" statements so the statements above are written with more general "a.s." condition.
Please correct me if I'm wrong. Thanks.
