The Difference Between Sigma Algebra and a Borel Measurable Function in Simple Terms

A sigma algebra is a collection of sets that satisfies three main criteria:

• The entire set, or the "whole universe" of possible outcomes, is part of this collection.
• If a set is in the collection, its opposite or complement is also in the collection.
• If you take any number of sets from the collection and group them together, that grouped set is also in the collection.

The purpose of a sigma algebra is to set the framework for which sets can be assigned a value, like probability.

On the other hand, Borel-measurable functions relate to how specific functions interact with our system of measuring sets. A function is considered Borel-measurable if, for any standard set from where the function outputs (like a certain range of numbers), the set of inputs that lead to those outputs belong to our predetermined measurable sets.

The main difference between the two is that a sigma algebra focuses on collections of sets and determining which can be measured. Borel-measurable functions are about ensuring a specific function behaves in a predictable and measurable manner when interacting with our measurement system.

Suppose we're analyzing the daily returns of a particular stock. Let's simplify by saying that our universe of possible outcomes consists of the stock going up, the stock going down, or the stock remaining flat on any given day. 

Our sigma algebra might be a collection of sets like:

• The set of all days (the entire set).
• The set of days where the stock goes up.
• The set of days where the stock goes down.
• The set of days where the stock remains flat.
• And combinations of the above (like days where the stock either goes up or remains flat).

The sigma algebra ensures that we can assign a probability to each of these events. For instance, we can determine the probability that the stock will go up tomorrow based on historical data.

Now, let's consider a function that represents a trading strategy. This function takes the stock's daily return as input and outputs a decision: "Buy", "Sell", or "Hold".

This function is Borel-measurable if, for every possible output (Buy, Sell, Hold), the set of daily returns that lead to that decision is in our sigma algebra. For example, if our function says "Buy" for days when the stock goes up and "Sell" for days when it goes down, the function is Borel-measurable since those sets of days are in our sigma algebra.

The concept ensures that our trading strategy is consistent with our system of measuring and predicting stock returns.

In essence, in quantitative finance, a sigma algebra helps determine which market scenarios or events we can analyze using probabilities, while Borel-measurable functions ensure that our strategies and models are consistent with this analytical framework.