What are the foundations of probability and how are they dependent upon a $\sigma$-field? I am reading Christopher D. Manning's Foundations of Statistical Natural Language Processing  which gives an introduction on Probability Theory where it talks about $\sigma$-fields. It says, 

The foundations of probability theory depend on the set of events $\mathscr{F}$ forming a $\sigma$-field". 

I understand the definition of a $\sigma$-field, but what are these foundations of probability theory, and how are these foundations dependent upon a $\sigma$-field?

Probability when there are only finitely many outcomes is a matter of counting. There are $36$ possible results from a roll of two dice and $6$ of them sum to $7$ so the probability of a sum of $7$ is $6/36$. You've measured the size of the set of outcomes that you are interested in.
It's harder to make rigorous sense of things when the set of possible results is infinite. What does it mean to choose two numbers at random in the interval $[1,6]$ and ask for their sum? Any particular pair, like $(1.3, \pi)$, will have probability $0$. 
You deal with this problem by replacing counting with integration. Unfortunately, the integration you learn in first year calculus ("Riemann integration") isn't powerful enough to derive all you need about probability. (It is enough to determine the probability that your two rolls total exactly $7$ is $0$, and to find the probability that it's at least $7$.) 
For the definitions and theorems of rigorous probability theory (those are the "foundations" you ask about) you need "Lebesgue integration". That requires first carefully specifying the sets that you are going to ask for the probabilities of - and not every set is allowed, for technical reasons without which you can't make the mathematics work the way you want. It turns out that the set of sets whose probability you are going to ask about carries the name "$\sigma$-field" or "sigma algebra". (It's not a field in the arithmetic sense.)
The essential point is that it's closed under countable set operations. That's what the "$\sigma$" says. Your text may not provide a formal definition - you may not need it for NLP applications.
