Yan's theorem

Notation:

${\displaystyle {\overline {\Omega }}}$ is the closure of a set ${\displaystyle \Omega }$.

${\displaystyle A-B=\{f-g:f\in A,\;g\in B\}}$.

${\displaystyle I_{A}}$ is the indicator function of ${\displaystyle A}$.

${\displaystyle q}$ is the conjugate index of ${\displaystyle p}$.

Let ${\displaystyle (\Omega ,{\mathcal {F}},P)}$ be a probability space, ${\displaystyle 1\leq p<+\infty }$ and ${\displaystyle B_{+}}$ be the space of non-negative and bounded random variables. Further let ${\displaystyle K\subseteq L^{p}(\Omega ,{\mathcal {F}},P)}$ be a convex subset and ${\displaystyle 0\in K}$.

Then the following three conditions are equivalent:

1. For all ${\displaystyle f\in L_{+}^{p}(\Omega ,{\mathcal {F}},P)}$ with ${\displaystyle f\neq 0}$ exists a constant ${\displaystyle c>0}$, such that ${\displaystyle cf\not \in {\overline {K-B_{+}}}}$.
2. For all ${\displaystyle A\in {\mathcal {F}}}$ with ${\displaystyle P(A)>0}$ exists a constant ${\displaystyle c>0}$, such that ${\displaystyle cI_{A}\not \in {\overline {K-B_{+}}}}$.
3. There exists a random variable ${\displaystyle Z\in L^{q}}$, such that ${\displaystyle Z>0}$ almost surely and

${\displaystyle \sup \limits _{Y\in K}\mathbb {E} [ZY]<+\infty }$.

• Yan, Jia-An (1980). "Caracterisation d' une Classe d'Ensembles Convexes de ${\displaystyle L^{1}}$ ou ${\displaystyle H^{1}}$". Séminaire de probabilités de Strasbourg. 14: 220–222.
• Freddy Delbaen and Walter Schachermayer: The Mathematics of Arbitrage (2005). Springer Finance