e start from an example. We would like to describe the notion
of almost sure (=almost everywhere) convergence of the random variables
to the random variable
on the set
with respect to a probability
.
The English statement would be the following: "There exists a subset
of the set
such that
and for any point
of the set
and any however small positive number
there exists an integer
depending on
and
such that for any integer
greater then
the following inequality holds:
."
The mathematical notation for such statements
is


(Almost sure convergence)

The
stands for "any",
is "there exists some", s.t. is "such that". Every variable
is bound by one of the operators
or
.
If the variable is not bound then the logical statement is meaningless.
Next, we describe a situation when
does not converge to
on
Such statement is the exact opposite to the statement above. The general
recipe is to permute
and
and replace the key relationship to the
opposite:
Again, the boundedness of every variable to
or
is important.
Let us now describe the notion of convergence in terms of sets. We start from
the set of all points
where
converges to
and represent it in terms of more elementary
sets:
Notably, we describe the
("any") using the intersection and describe the
("exists") using the union of the sets.
Let
be the total set. We would like to describe the equivalent of
negation:
We note the following property of set
operations


(Intersection property)

Hence,
We use another property of set operations:


(Union property)

Thus,
Note how this result matches the negative statement obtained via the logical
operations.
There will be numerous applications of this technique in the present chapter.
