Stochastic Process

Laws of large numbers (LLN's) specify what 'become deterministic' means.
They only operate within the extended model, in other words, laws of large numbers don't apply to the real world

Let's talk about Markov bounds, chebychev bounds and chernoff bounds. what it says is if Y is a non-negative R.V. with an expectation E[Y], then for any real y greater than 0, the probability that Y is greater than or equal to y is less than or equal to the expected value of Y divided by y. The proof of it is by picture.

Properties of Limit Inferior & Limit Superior

Lebesgue's Integrability Condition

Lebesgue's integrability condition, aka Lebesgue's criterion for Riemann integrability, Riemann--Lebesgue theorem

A bounded function on a compact interval [a, b] is Riemann integrable if and only if it is continuous a.e.

The criterion has nothing to do with the Lebesgue integral. It is due to Lebesgue and uses his measure zero, but makes use of neither Lebesgue's general measure or integral.