2

I have studied measure theory and probability theory before, and we defined the expectation of a random variable $X$ with respect to some probability space $(\Omega,\mathcal{F},\mathbb{P})$ as the Lesbesgue integral of $X$ w.r.t. $\mathbb{P}$, that is, $$\mathbb{E}[X]=\int_{\Omega}Xd\mathbb{P}=(\lambda\otimes\mathbb{P}) \ \text{hyp}(X),$$ where $\text{hyp}(X)$ denotes the hypograph of $X$. This made sense to me, and we proved results about what happens when the random variable is discrete, admits a pdf, etc.

Now, I am reading a statistics book which isn't theoretical, which defines the expectation of $X$ in terms of the Riemann-Stieltjes integral: that is, letting $F(x)=\mathbb{P}[X\leq x]$, we have $$\mathbb{E}[X]=\int_{-\infty}^{\infty}XdF(x),$$

the Riemann-Stieltjes integral.

My question: Clearly these two must be equal, but (just looking in terms of probability theory here) does it hold for all random variables (e.g. functions of $X$) and all probability spaces? When would we prefer to use one over the other?

I suspect my book introduced the Riemann-Stieltjes version to me to avoid a bunch of measure theory, but if I am comfortable with measure theory, would I benefit from considering the measure theoretic definition of the expectation over the Riemann-Stieltjes definition?

jacob
  • 602
  • "but if I am comfortable with measure theory, would I benefit from considering the measure theoretic definition of the expectation over the Riemann-Stieltjes definition?" That's at least what I chose for myself. Whenever I see these Riemann-Stieltjes guys, I translate it into measure stuff in my head. – amsmath Aug 01 '21 at 17:14

1 Answers1

2

Riemann-Stieltjes is just fine for most elementary applications of probability theory. However, as a general formulation of expectation you need to use Lebesgue integration to handle the more “exotic” sample spaces one can construct.

As a simple (and well-worn) example — define a random variable over the Reals as such:

$$X(\omega): \mathbb{R} \to \{0,1\}, \omega \mapsto \mathbf{1}_{\mathbb{Q}}(\omega)$$

Given a probability space $(\mathbb{R}, \mathcal{B}(\mathbb{R}),\mathbb{P})$, what is $E_{\mathbb{P}}[X]$?

This function is not Riemann integrable and so you cannot use it to calculate the expectation, but under Lebesgue integration we see that it is always $0$.