Sure Convergence
To say that the sequence or random variables (Xn) defined over the same probability space (i.e., a random process) converges surely or everywhere or pointwise towards X means
where Ω is the sample space of the underlying probability space over which the random variables are defined.
This is the notion of pointwise convergence of sequence functions extended to sequence of random variables. (Note that random variables themselves are functions).
Sure convergence of a random variable implies all the other kinds of convergence stated above, but there is no payoff in probability theory by using sure convergence compared to using almost sure convergence. The difference between the two only exists on sets with probability zero. This is why the concept of sure convergence of random variables is very rarely used.
Read more about this topic: Convergence Of Random Variables