Question

I was having difficulty in understanding the difference between convergence in probability and almost sure convergence, so I decided to try to reduce them to some sort of canonical form.

• Convergence in probability: ${\lim}_{n \to \infty } Pr(|X_n-X| \ge e)=0$
• Almost sure convergence: $Pr({\lim}_{n \to \infty} X_n=X)=1$

After playing around with the figures, I got the following results.

• Convergence in probability: $\forall e, d, n>N(e,d): dif_x \ge e \text{ with } p < d $
• Almost sure convergence: ($\forall e, n>N(e): dif_x < e) \text{ with } p=1$
• Alternative form: ($\forall e, n>N(e): dif_x \ge e) \text{ with } p=0$

A few notes:

1. Here $dif_x$ means how far about points at this location are from the limit
2. $N(e,d)$ simply says that we can find a suitable value of N so that this holds which depends on e and d
3. The differences between the two types seem more obvious in this form

So, my questions are:

1. Is this correct?
2. Have reductions into this kind of form been studied? If so, where can I learn more about this?

Answer 1

1. Maybe I didn't get the notation, but if you suppose, that $\operatorname{dif}_x$ is just $|T_n(x) - T(x)|$, then 1, first and second reformulation, is not correct: suppose all functions on $[0, 1]$, $T \equiv 0$ and $T_n(x) = \chi_{[0, \frac1n]}$ — the characteristic function of $[0, \frac1n]$, then for $d = 1$ and for all $n$ on the set of positive measure ($[0, \frac1n]$) the difference is not smaller then $1$. Third reformulation is correct.

2. Dyachenko and Ulyanov in their book 'Measure and integration' (unfortunately only in russian or spanish) used similar notions: $F_{k, m} = \left[x \in X: |T_k(x) - T(x)| > \frac1m \right]$: then if we denote $E = \left[x \in X : T_k(x) \to T(x) \text{as} k \to \infty \right]$ we will have $$X \setminus E = \cup_{m=1}^\infty \cap_{n=1}^\infty \cup_{k=n}^\infty F_{k, m}$$