Independent symmetric 3-valued random variables in $L_p$

Question

Consider the following excerpt from this paper:

Given $1<p<2$, $0<w\leq 1$ and $n\in\mathbb{N}$, we fix once and for all a sequence $f_j^{(n)}=f_{p,w,j}^{(n)}$, $1\leq j\leq n$, of independent, symmetric, 3-valued random variables with $\|f_j^{(n)}\|_{L_p}=1$ and $\|f_j^{(n)}\|_{L_2}=\frac{1}{w}$ for $1\leq j\leq n$. We then define $F_{p,w}^{(n)}$ to be the subspace $\text{span}\{f_j^{(n)}:1\leq j\leq n\}$ of $L_p$. ... Since the $f_j^{(n)}$ are 3-valued, $F_{p,w}^{(n)}$ is a subspace of the span of indicator functions of $3^n$ pairwise disjoint sets. Thus, we can and will think of $F_{p,w}^{(n)}$ as a subspace of $\ell_p^{k_n}$, where $k_n=3^n$.

Two things about this, I don't understand.

Question 1. How do we know that such $f_j^{(n)}$ exist?

Question 2. I do not understand the last sentence above, about $F_{p,w}^{(n)}$ being viewed as a subspace of $\ell_p^{3^n}$. Don't the pairwise disjoint sets they mention all need to have the same measure? But this is impossible since the space $[0,1]$ has finite measure. Otherwise, the embeddings will not be uniform.

EDIT: $\|f_j^{(n)}\|_{L_p}=1$ means that $(\int_0^1|f_j^{(n)}(x)|^pdx)^{1/p}=1$, and $\|f_j^{(n)}\|_{L_2}=\frac{1}{w}$ means that $(\int_0^1|f_j^{(n)}(x)|^2dx)^{1/2}=\frac{1}{w}$. I'm not sure what symmetric means in this context, but I think it means that $f_j^{(n)}(\frac{1}{2}-\delta)=f_j^{(n)}(\frac{1}{2}+\delta)$ for all $\delta\in[0,\frac{1}{2}]$.

Answer 1

A random variable $X$ is symmetric if for each Borel set $B$ we have $\mathsf{P}(X\in B) = \mathsf{P}(-X\in B)$.

Regarding your first question, we are looking for a measurable simple function $$f=-\mathbf{1}_{A_1} + 0\cdot \mathbf{1}_{A_2}+ \mathbf{1}_{A_3}$$ (once we have one, we have a sequence of independent copies) such that

• $\mu(A_1)=\mu(A_3)$

• $\mu(A_1)^p + \mu(A_3)^p = 1$

• $\sqrt{\mu(A_1)^2 + \mu(A_3)^2}=\frac{1}{w}$

• $A_1 \cup A_2 \cup A_3 = [0,1]$.

Can you solve this system of equations for Borel sets $A_1, A_2, A_3$ that are pair-wise disjoint?

The very idea of the proof relies on a theorem of Rosenthal which asserts that any sequence of independent 3-valued random variables in $L_p$ ($p\in (2,\infty)$) spans a complemented subspace of $L_p$ isomorphic to Rosenthal's space $X_p$; see his paper:

H. P. Rosenthal, On the subspaces of $L_p$ ($p > 2$) spanned by sequences of independent random variables. Israel J. Math. 8 (1970), 273–303.

No, the measures of supports need not be the same; the resulting finite-dimensional space you get is actually a weighted $\ell_p^k$-space, hence (uniformly) isomorphic to $\ell_p^k$.