Cardinality of a set of random variables (measurable functions): $|L_0|=|L_p|=|L_\infty|$ for any $p >0$?

Question

How can we compare the cardinalities of different sets of random variables on a given probability space $(\Omega, \mathcal F, P)$.

Is $|L_1|>|L_\infty|$, while $L_\infty$ is dense in $L_1$?

When does the following $$|L_p|=|L_\infty|$$ hold for all $p\ge1$?

How large can be the set $L_0$ of all random variables compared to $L_\infty$?

How much are we lucky if a given distribution has a finite mean or finite variance?

How restrictive is if a statement only holds for random variables with mgf?

How are these cardinalities connected to the cardinality of $\Omega$,

See Wikipedia for definition of $L_p$ spaces. Here, $L_0$ denotes the set of all random variables (Borel measurable functions) defined on the probability space.

Specifically interested in two cases of $(\Omega, 2^\Omega)$ where $\Omega$ is a finite set and $(\mathbb R, \mathcal B(\mathbb R)).$ For the latter, from here, we know that $$|L_0|=|\mathbb R|,$$

whereas the cardinality of the set of all Lebesgue measurable functions is $|2^\mathbb R|$ (which shows how small is the set of random variables, but note that they have the same cardinality of $|\mathbb R|$ considering equivalence classes). From the above, I think we have $$|L_{\frac{1}{p}}|=|L_p|=|\mathbb R|$$ for all $p\ge1$ as $$|\mathbb R|=|L_0|\ge|L_{\frac{1}{p}}|\ge|L_p|\ge|L_\infty|\ge|\mathbb R|.$$ On the other hand, when $\Omega$ is finite, we have

$$|L_0|=|L_p|=|L_{\frac{1}{p}}|=|L_\infty|$$

for any $p\ge1$.

Can we generalize these two observations as follows?

1- For any given $(\Omega, \mathcal F, P)$, we always have $$|L_0|=|L_{\frac{1}{p}}|=|L_p|=|L_\infty|$$ for any $p\ge 1$.

If yes, then

2- How can we define a bijection (one-to-one correspondence) for each pair of these spaces?

In question 1, I also guess considering equivalence classes does not change the result.

---

PS: Related Stack Exchange questions on sets with $|\mathbb R|$ cardinality:

Cardinality of set of real continuous functions

Cardinality of the borel measurable functions

Cardinality of the set of entire functions

Cardinality of the set of Lebesgue measurable functions under equivalence

The cardinality of the Riemann integrable functions is $|2^\mathbb R|.$

Cardinality of the set of Riemann integrable functions on [0,1]

Answer 1

Fix $(\Omega, \mathcal{F}, P)$ as a probability space. A random variable is a Borel measurable function $X:\Omega\rightarrow\mathbb{R}$. Define:

\begin{align*} U &= \{\mbox{Random variables $X:\Omega\rightarrow \mathbb{R}$}\}\\ B &= \{\mbox{Random variables $X:\Omega\rightarrow (-\pi/2, \pi/2)$} \} \end{align*}

Claim: $|B|=|U|$.

Proof: Note that if $X \in U$ then $\arctan(X)$ is a random variable (since it is a continuous function of $X$) and $\arctan(X) \in B$. Define the function $f:U\rightarrow B$ given by $$ f(X) = \arctan(X) \quad \forall X \in U$$ This function $f$ is surjective because if $Y\in B$ then $\tan(Y) \in U$ and $f(\tan(Y))=Y$. This function $f$ is injective since if $X_1, X_2$ are two random variables in $U$ that satisfy $f(X_1)=f(X_2)$, then $\arctan(X_1)=\arctan(X_2)$ and so $X_1=X_2$. Thus, $f$ is a bijection between $U$ and $B$. $\Box$

---

• In the special case when $\mathcal{F}=Pow(\Omega)$ then all functions $X:\Omega\rightarrow\mathbb{R}$ are random variables and $|U|=|\mathbb{R}^{\Omega}|$.

• In the special case when $\Omega$ is finite or countably infinite and $\mathcal{F}=Pow(\Omega)$ then $|U|=|\mathbb{R}|$, since $$ |\mathbb{R}|\leq |\mathbb{R}^{\Omega}|\leq |\mathbb{R}^{\mathbb{N}}|=|\mathbb{R}|$$