TITLE: Expectation Functional in Lebesgue and Riemann Terms – Looking for a clarification
QUESTION [5 upvotes]: Here there is a really central problem I am having self-studying probability theory, that concerns the relation between the definition of expectation in Lebesgue terms and in Riemann terms. I will truly appreciate any feedback because I feel this is really at the very core of the all theory.
Summing up, I have basically two problems (that are closed link):

I really don't see why the following two expressions
$$ \mathbb{E}(X) :=\int_\Omega X d P  \hspace{2cm}(L)$$
and
$$ \mathbb{E}(X) := \int_{-\infty}^{\infty} t f(t) dt \hspace{2cm}(R)$$
are equivalent.
[Of course, here I am referring to the case of $X$ continuous r.v.]

I am not able to let coexist the idea that the Lebesgue Integral gives us
a. a number that is the area under a certain curve, and
b. that basically the same number represents the average value that is taken by this curve.


Is there somebody who can enlighten me?
I would say that the second question is really the first, but disguised.
As always, any feedback is most welcome.
Thank you for your time.

REPLY [2 votes]: (L) is the definition. (R) is not always true.

Let $X$ be a $(\mathbb R, \mathscr B (\mathbb R))$-valued random variable on a probability space $(\Omega, \mathscr F, \mathbb P)$. Its law is given by $\mathcal L_X(B) = P(X \in B)$ where $B \in \mathscr B (\mathbb R)$
By change of variable theorem we have the Lebesgue integral:
$$E[X] = \int_{\mathbb R} t d\mathcal L_X(t)$$
If $\mathbb P$ is Lebesgue measure, we have the Stieltjes integral:
$$E[X] = \int_{\mathbb R} t dF_X(t)$$
If $X$ is absolutely continuous, it has a pdf and then we have the Riemann integral:
$$E[X] = \int_{\mathbb R} t f_X(t) dt$$

The way the Riemann integral is defined is through areas of rectangles that approximate an area under a curve.

The way the Lebesgue integral is defined is through the standard machine: indicator functions, simple functions, nonnegative functions and then integrable functions.
It's not always supposed to give an area under a curve because the integrands of Lebesgue integrals are not necessarily continuous. I wouldn't say the Lebesgue integral is supposed to assign an area under a curve. It seems more like a way to assign a general 'measure' rather than merely an 'area'
Say for example we have $$E[X] = \int_{\Omega} X dP$$
where $X$ is a discrete random variable. It wouldn't make sense to compute the 'area' under $X$ because it would just be zero right? However, it could make sense to compute the 'measure' under $X$. Try to look back at your notes or look online as to why we have Lebesgue integration in the first place.