Linear and almost linear functions on a~measurable Hilbert space
Teoriâ veroâtnostej i ee primeneniâ, Tome 23 (1978) no. 2, pp. 397-402
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Let $X$ be a separable Hilbert space, $\mathfrak B$ be the Borel $\sigma$-algebra in $X$, and ($\mu$ be a probability measure on $\mathfrak B$. A function $\varphi(x)$ is called a $\mu$-measurable linear function if it is the limit in $\mu$ of a sequence of continuous linear functions. A function $\varphi(x)$ is called an almost linear function, if it is $\mathfrak B$-measurable and there exists a linear $\mathfrak B$-measurable manifold $L\subset X$ such that $\mu(L)=1$ and $\varphi(x)$ is linear on $L$.
We investigate the class of all linear functions and (in the case of quasiinvariant measure) the class of all almost linear functions.
@article{TVP_1978_23_2_a14,
author = {A. V. Skorohod},
title = {Linear and almost linear functions on a~measurable {Hilbert} space},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {397--402},
publisher = {mathdoc},
volume = {23},
number = {2},
year = {1978},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_1978_23_2_a14/}
}
A. V. Skorohod. Linear and almost linear functions on a~measurable Hilbert space. Teoriâ veroâtnostej i ee primeneniâ, Tome 23 (1978) no. 2, pp. 397-402. http://geodesic.mathdoc.fr/item/TVP_1978_23_2_a14/