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Events

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Probability Space

(&Omega., @F, P) is a ~{probability triple} if it is a measure space and P(&Omega.) = 1.")
P is called the ~{probability measure} on the ~{sample space} &Omega.. _ A point &omega. &in. &Omega. is called a ~{sample point}.
An ~{event} is an element F of the &sigma.-algebra @F _ - the family of events. I.e. an event is an @F-measurable subset of &Omega..

Almost Surely (a.s.)

The ~{truth set} of a statement @S about outcomes is the set

Tr(@S) #:= \{ &omega. | @S(&omega.) is true \} A statement @S is true ~{almost surely} _ if _ P(Tr(@S)) = 1

Proposition
If F_{~n} &in. @F, _ ~n &in. &naturals., _ and P(F_{~n}) = 1, _ &forall. ~n, _ then;

P rndb{intersect{F_{~n},~n,}} _ = _ 1

Lim sup, lim inf, etc., of Real Numbers

~{x_n} &in. &reals., _ recall that the ~#{least upper bound} or ~#{supremum} _ ~s = sup \{ ~{x_n} \} _ if

~s &ge. ~{x_n}, _ &forall. ~n, _ and if _ ~t &ge. ~{x_n}, &forall. ~n, _ then ~t &ge. ~s.

inf \{ ~{x_n} \} is defined in an analogous manner. Then we have:

lim sup ~{x_n} _ #:= _ inf_{~m} \{ sup_{~n&ge.~m} ~{x_n} \} _ = _ &downarrow.lim_{~m} \{ sup_{~n&ge.~m} ~{x_n} \} _ &in. _ [&minus.&infty., &infty.]

where &downarrow.lim indicates the limit of a decreasing sequence. _ [Obviously _ sup_{~n&ge.~m} ~{x_n} _ &ge. _ sup_{~n&ge.~m+1} ~{x_n}.]

Similarly:

lim inf ~{x_n} _ #:= _ sup_{~m} \{ inf_{~n&ge.~m} ~{x_n} \} _ = _ &uparrow.lim_{~m} \{ inf_{~n&ge.~m} ~{x_n} \} _ &in. _ [&minus.&infty., &infty.]

We have:

~{x_n}converges in [&minus.&infty., &infty.] _ _ <=> _ _ lim inf ~{x_n} _ = _ lim sup ~{x_n}

In which case: _ _ _ _ _ lim ~{x_n} _ = _ lim inf ~{x_n} _ = _ lim sup ~{x_n}

Note that

  • ~t > lim sup ~{x_n} _ then _ ~{x_n} < ~t _ ~{eventually}
    i.e. for all sufficiently large ~n.
  • ~t < lim sup ~{x_n} _ then _ ~{x_n} > ~t _ ~{infinitely often}
    i.e. for infinitely many ~n.

Lim sup, lim inf, etc., of Events

Let ( A_{~n}#: ~n &in. &naturals.) be a collection of (sub)sets.
If _ A = &union. A_{~n}, _ then _ A_{~n} &subseteq. A, &forall. ~n, _ and _ A_{~n} &subseteq. B, &forall. ~n, _ &imply. A &subseteq. B
If _ C = &intersect. A_{~n}, _ then _ C &subseteq. A_{~n}, &forall. ~n, _ and _ D &subseteq. A_{~n}, &forall. ~n, _ &imply. D &subseteq. C
So by analogy with the real numbers:

sup A_{~n} _ _ #:= _ _ union{A_{~n},~n,}

inf A_{~n} _ _ #:= _ _ intersect{A_{~n},~n,}

Now suppose ( E_{~n}#: ~n &in. &naturals.) is a sequence of events:

Infinitely Often

( E_{~n}, i.o.) _ _ #:= _ _ ( E_{~n}, infinitely often)

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ #:= _ _ lim sup E_{~n} _ _ #:= _ _ intersect{union{E_{~n},~n&ge.~m,},~m,}

Eventually

( E_{~n}, ev.) _ _ #:= _ _ ( E_{~n}, eventually)

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ #:= _ _ lim inf E_{~n} _ _ #:= _ _ union{intersect{E_{~n},~n&ge.~m,},~m,} Note that _ _ ( E_{~n}, ev.)^{~c} _ = _ ( E_{~n}^{~c}, i.o.)

Reverse Fatou Lemma

This depends on the finiteness of P.

P( lim sup E_{~n}) _ _ _ _ >= _ _ _ _ lim sup P( E_{~n})

Borel-Cantelli, First Lemma (BC1)

Let ( E_{~n}#: ~n &in. &naturals.) be a sequence of events such that _ &sum._{~n} P( E_{~n}) < &infty.. Then

P( lim sup E_{~n}) _ = _ P( E_{~n}, i.o.) _ = _ 0

Fatou's Lemma for sets

True for all measure spaces:

P( lim inf E_{~n}) _ _ _ _ =< _ _ _ _ lim inf P( E_{~n})