On the closedness of the sum of marginal subspaces on $[a,b)$
Volume 280 / 2025
Abstract
Let $(\Omega ,\mathcal {F},\mu )$ be a probability space and $p\in [1,+\infty )$. For every sub-$\sigma $-algebra $\mathcal {A}$ of $\mathcal {F}$ we define the marginal subspace of $L^p(\Omega ,\mathcal {F},\mu )$ corresponding to $\mathcal {A}$, $L^p(\mathcal {A})$, to consist of the equivalence classes of $\mathcal {A}$-measurable $p$-integrable random variables; it is a closed subspace in $L^p(\Omega ,\mathcal {F},\mu )$.
Let $\Omega =[a,b)$, where $-\infty \lt a \lt b\le +\infty $. Denote by $\mathcal {B}([a,b))$ the Borel $\sigma $-algebra on $[a,b)$. Let $\mu $ be a probability measure on $\mathcal {B}([a,b))$ and $p\in [1,+\infty )$. For a sequence of points $\pi =\{a_1,a_2,\ldots \}$, where $a \lt a_1 \lt a_2 \lt \cdots $ and $a_k\to b$ as $k\to \infty $, define a partition of $[a,b)$ by ${\rm part}(\pi )=\{[a,a_1),[a_1,a_2),\ldots \}$. Let $\sigma a(\pi )$ be the $\sigma $-algebra generated by ${\rm part}(\pi )$. The marginal subspace $L^p(\sigma a(\pi ))$ of $L^p(\mathcal {B}([a,b)))$ consists of (the equivalence classes of) $p$-integrable random variables which are constant on each element of ${\rm part}(\pi )$. Let $\pi _1,\ldots ,\pi _n$ be sequences of points of $[a,b)$. We study the following question: when is $L^p(\sigma a(\pi _1))+\cdots +L^p(\sigma a(\pi _n))$ closed in $L^p(\mathcal {B}([a,b)))$? We establish a relation between closedness of such sums and “fast decreasing” of tails of the measure $\mu $. We also outline an application of our result to the $L^2$-version of additive modeling.
