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Sup /sci/,
I'm working on a proba exercice, but I wonder one thing :
If [math] ( \Omega, \mathcal O, \mu ) [/math] is a measurable space, [math] \{ X_n \}_n [/math] a sequence of random functions, is :
[eqn] \{ \sup_n X_n = 0 \} = \bigcup_n \{ X_n = 0 \} [/eqn]
and
[eqn] \{ \inf_n X_n = 0 \} = \bigcap_n \{ X_n = 0 \} [/eqn]
Moreover, have we, for any borelian [math]\mathcal B [/math] :
[eqn] \{ \sup_n X_n \in \mathcal B \} = \bigcup_n \{ X_n \in \mathcal B \} [/eqn]
and
[eqn] \{ \inf_n X_n \in B \} = \bigcap_n \{ X_n \in B \} [/eqn]
I know that :
[eqn] \{ \lim_n \sup X_n \in B \} = \bigcap_n \bigcup_ {m > n} \{ X_m \in B \} [/eqn] but I do not know how to prove it.
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Moreover, I'm NOT asking some help for my homework, my homework are completely different, so fuck you normiefag who wanted to call the mods.
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>>8387315
hey man, seems that you actually know math!!!
i study math as well (undergrad), but I hate probability so I'm not even gonna look at this (I'm a Pure guy... aldgebra and shit).
I think you spent too much time asking a question at a place where most users just psudo-science all day long and are mostly morons...
There are math forums out there...
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>>8387334
Probability is literally pure math. Hell, even statistics can be done as 'pure' math but its more of a stretch.
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>>8387334
I can't tell whether this is sincere or extremely artful bait
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>>8387358
>Probability is literally pure math
hahahahahaha
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>>8387376
Go ahead and try to show otherwise.
>inb4 posting non-rigorous applied probability
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>>8387358
>Probability is literally pure math.
no
eat shit you brainlet
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>>8387465
Probability Theory is pure math. You can make it as applied as you want but at the end of the day the topic is pure.
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>>8387315
I actually thought that was the definition of limsup (and liminf being the opposite).
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