$f_n(x)\rightarrow f(x)$ almost everywhere

measure-theoryreal-analysis

$$\lim_{n\rightarrow\infty} f_n(x)=f(x)\space\space a.e.\text{for all x in E}$$

My Question:

  1. How do you interpret this? As you get to the tail of the function sequence, each tail sequence and the function disagree only on negligible sets?

  2. Is there a name for this type of convergence other than what we call almost everywhere convergence?

  3. Is there any relationship between the a.e. convergence and pointwise convergence or uniform convergence?

These are discussed in the context of constructing the Lebesgue integration theory.

Reference:
$\textit{Real Analysis: Measure Theory, Integration, and Hilbert Spaces}$. Elias M. Stein, Rami Shakarchi. Princeton University Press, 2009.

Best Answer

The definition says that the set $\{x\in E: f_n(x)$ does not converge to $f(x)\}$ has measure zero. Obviously this is something a bit weaker than pointwise convergence. (pointwise convergence is when the set I defined is empty).

As for names: in probability theory this type of convergence is also called "almost sure convergence". The intuition there is that if you pick a point then with probability $1$ it is a point where the sequence converges to the limit random variable.

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