PomeDefinition 16.29.6.949E1D79FF19 An envelope system over $\theta $, with parameters $a<c$, is a family
\[ \boldsymbol \Gamma =(\Gamma ^\eta )_{\eta \in \mathscr H(X)}, \qquad \Gamma ^\eta \in \mathscr T_{a,c}(\theta +\eta ), \]
such that
101FB6DC67C6
16.29.7
\begin{equation} \Gamma ^\eta =P_{\theta +\eta }[\Gamma ^{\eta '}]_{\mathcal I} \quad \hbox{whenever }\eta \geq \eta '. \end{equation}
Write $\mathscr S_{a,c}(\theta )$ for the set of these systems, ordered componentwise. For $\Gamma \in \mathscr T_{a,c}(\theta )$, set
\[ \mathsf E(\Gamma )^\eta =P_{\theta +\eta }[\Gamma ]_{\mathcal I} \qquad (\eta \in \mathscr H(X)). \]
A system is realized in the background $\theta $ if it equals $\mathsf E(\Gamma )$ for some $\Gamma \in \mathscr T_{a,c}(\theta )$.
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.