Let $\scr{C}$ be a collection of subsets. Prove that if $A \in \sigma(\scr{C})$ (sigma algebra generated by $\scr{C}$), then there exists a countable subcollection $\scr{C}_A$ of $\scr{C}$ such that $A \in \sigma(\scr{C}_A)$  If $\scr{C}$ is finite, I think the result is immediate. Assume $\scr{C}$ is infinite. Then I know that $\sigma(\scr{C})$ is uncountable (infinite sigma algebra is always uncountable). That all I know. I think that it is impossible to have a clear form of a set belonging to a given infinite sigma algebra. So it is really no clue how to do it. I try contradiction, but no further development for the case when $\scr{C}$ is uncountable.  Any suggestion please ?
Given a collection of sets $\mathcal{C}$ and $E$ an element in the $\sigma$-algebra generated by $\mathcal{C}$, how do I show that $\exists$ a countable subcollection $\mathcal{C_0} \subset \mathcal{C}$ such that $E$ is an element of the $\sigma$-algebra, $\mathcal{A}$ generated by $\mathcal{C_0}$?    The hint says to let $H$ be the union of all $\sigma$-algebras generated by countable subsets of $\mathcal{C}$....although I don't know why.
How do I bulkify queries for directly or indirectly related records? This is a canonical question and answer developed by the community to help address common questions. If you've been directed here, or your question has been closed as a duplicate, please look through the resources here and use them to shape more specific questions. To browse all canonical questions and answers, navigate to the  tag. This question and answer is owned by the community! If you can improve it and make it more useful, you're welcome to edit it. I have a collection of records, like a List&lt;sObject&gt; or the Trigger.new context variable. I need to query other records that are related to these records either directly or indirectly, and I want to use the related data to make further updates. When I write a query in a loop, like for (Account a : Trigger.new) {     Account parent = [SELECT Name FROM Account WHERE Id = :a.ParentId];     // ... }  I keep hitting Salesforce query limits, and I know this is not best practice. What can I do to avoid this problem and bulkify my code? Transit visa requirements at Narita Airport Why is surface tension parallel to the interface? A text says:     The surface tension of a liquid results from an imbalance of intermolecular attractive forces, the cohesive forces between molecules:   A molecule in the bulk liquid experiences cohesive forces with other molecules in all directions.   A molecule at the surface of a liquid experiences only net inward cohesive forces   If this is the case: Why is surface tension considered or experienced as a force parallel to the interface while it is so obvious that it must be perpendicular to it? Usage of the word "meet" in the context of a phone call/video conference My Logitech G5 Laser mouse has 9 buttons but it's SetPoint software only allows 5 of them to be used as "mouse buttons" and the remaining 4 can only be assigned special windows functions (Forward, Back, Cruise Up, etc) or Keystroke Assignments.  Many of the games I play will take up to 10 "MOUSE BUTTON" inputs for in-game actions but wont recognize the remaining 4 buttons even when assigned keystroke assignments.  Is there a way to make these last 4 buttons "Generic Buttons" as well as buttons 3,4,and 5? (1 is left click and 2 is right click).  My other question is if the above is not possible, maybe I can use AutoHotkey to remap keystrokes as assigned by my SetPoint software to mouse button commands?  If so how? We are beginning to go down the path of mobile browser support for an enterprise e-commerce webapp (Java/Servlet based).  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