Measure theory in quantitative finance

When I read up on stochastic modeling, the use of "measure" comes up a lot. So far I just read the word "measure" as "probabilities" or "distribution" and was able to get away with it when trying to understand informally the concept and results of various models (Black, Dupire, Hull White..), at a basic level.

I must admit that I was not able to follow the rigorous proofs of every step in their derivation.

Can anyone tell me what is the significance of measure theory in stochastic modeling?

I do understand that the definition of a probability measure is more general and rigorous than just saying "probabilities". But apart from a rigorous definition, was there some useful results from measure theory that are often used in stochastic modeling?

Thanks very much and pardon my ignorance.

• The Borel-Cantelli Lemma, the Kolmogorov Zero One Law, the various forms of stochastic convergence cannot really be formulated or proven except with Measure Theory. – noob2 Aug 2 '18 at 20:20
• Measure theory does not just "add rigour", it provides new results and tools for dealing with infinite sets and infinite sequences, etc. in aprobaility context. This was is a great advance in Probability Theory for which we can thank academicain A. N. Kolmogorov. – noob2 Aug 2 '18 at 20:25
• OTOH if you just trade options all day long you probably don't need it – noob2 Aug 3 '18 at 16:56

Measure theory helps us overcome some of the drawbacks of constructing measures (measure of probability when ranged at $$[0,1]$$). Classic probability theory is effective for probability models whose sample space $$\Omega$$ is a finite or countable set ($$P: 2^{\Omega} \to [0,1]$$). But for uncountable sets, such as $$\mathbb{R}$$ (which is uncountable) the construction falls short. Measure theoretic probability, thanks to H.Lebesgue and C. Carathéodory enables measure construction on uncountable sets. Definition of a measure $$\mu$$, requires firstly the definition of the outer measure $$\mu ^*.$$ An outer measure is a mapping $$\mu^*:2^{\Omega} \to [0,+ \infty)$$:

1) $$\mu^*(\emptyset)=0$$

2) if $$A \subset B$$, then, $$\mu^*(A) \leq \mu^*(B)$$

3) if $$\{A_i\} \in 2^{\Omega}$$, a sequence of events, then $$\mu^*(\bigcup A_i) \le \sum_i \mu^*(A_i)$$

We would like to define a subset $$\mathcal{F} \subset 2^{\Omega},$$ such that 3) becomes an equality (This property is called countable additivity). Using Carathéodory's condition we are able to construct such a set $$\mathcal{F}$$, which is called a $$\sigma-$$algebra. A measure is $$\mu$$ is just the outer measure restricted in $$\mathcal{F}$$, $$\mu=\mu^{*} \mid_\mathcal{F}$$.The next step is to define the smallest $$\sigma-$$algebra, which is the intersection of all $$\sigma-$$algebras. This set is the Borel algebra. We continue on this manner.

Measure theory provides more robust methods for integration in measureable functions, almost everywhere convergence, Fatou's Lemma, Borel Cantelli Lemma and so on.

Later Black-Scholes was reformulated in terms of stochastic differential equations. So instead of assuming continuous trading you assume that stock (log-)prices behave like Brownian motion $$d\log(S) = \mu + \sigma dW_t$$ (which is more or less the same because Brownian motion is the continuous-time limit of all random walks) with drift. But to specify a drift parameter is to make an assumption about the future behavior of stocks, which of course doesn't work.
Everything in "naïve probability theory" is implicitly defined in terms of an "ambient" measure $$\mathbb P$$ that in finance is called the "historical" or "physical" probability measure. It's effectively the probability that holds in the real world. But as it turns out (Girsanov's theorem), there is a probability measure $$\mathbb Q$$, the "risk-free measure" where you can change the drift of your process from its value under the physical measure to something else. With some additional assumptions and math that defines a portfolio and arbitrage opportunities, you get that you should be using the risk-free interest rate as the drift under $$\mathbb Q$$. Then you solve the equation without making assumptions about the physical drift.