Exchange of Order of Supremum Operators

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $\family {a_i}_{i \mathop \in I}$ be a family of elements of the non-negative real numbers $\R_{\ge 0}$ indexed by $I$.

Let $\family {b_j}_{j \mathop \in J}$ be a family of elements of the non-negative real numbers $\R_{\ge 0}$ indexed by $J$.

Let $\map R i$ and $\map S j$ be propositional functions of $i \in I$, $j \in J$.


Let $\ds \sup_{\map R i} a_i$ and $\ds \sup_{\map S j} b_j$ be the indexed suprema on $\family {a_i}$ and $\family {b_j}$ respectively.


Then:

$\ds \sup_{\map R i} \paren {\sup_{\map S j} a_{i j} } = \sup_{\map S j} \paren {\sup_{\map R i} a_{i j} }$


Proof


This theorem requires a proof.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.



Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Exchange_of_Order_of_Supremum_Operators&oldid=533159"