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I am trying to implement a Google Analytics user-level custom dimension on a site.

I have read here that the code to do this looks something like this:

ga('set', 'dimension2', value );

If I include that script in page X, will the hit for page X's page-view include that dimension/value? Or will it only apply to future hits that that user initiates?

Since you're setting a user level scoped CD, it will apply to all previous hits in the session as well.

I'd recommend you do the ga('set') before you perform ga('send'), this way you utilize the pageview hit to set this CD, not the next hit.

Edit:
Your regular implementation looks like this:

<!-- Google Analytics -->
<script>
(function(i,s,o,g,r,a,m)function()).push(arguments),i[r].l=1*new Date();a=s.createElement(o),
m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;m.parentNode.insertBefore(a,m)
)(window,document,'script','https://www.google-analytics.com/analytics.js','ga');

ga('create', 'UA-XXXXX-Y', 'auto');

ga('set', ...); //do you sets here

ga('send', 'pageview'); //this is the send
</script>
<!-- End Google Analytics -->

Alternatively, you can just run this and just do "set" and "send" separately, though you'd want to execute the set/send very close to when you load the page:

 <!-- Google Analytics -->
 <script>
 (function(i,s,o,g,r,a,m)i['GoogleAnalyticsObject']=r;i[r]=i[r])(window,document,'script','https://www.google-analytics.com/analytics.js','ga');

 **//removed ga(send) stuff**

 </script>
 <!-- End Google Analytics -->

To verify, use dev tools, look at your networks tab. Search/filter for "collect" you'll see the calls made to GA. Look for the parameters for the call, specificall cd1, cd2, cd3 etc...look for the cd# that's the same index as the cd you've set. Verify the data there is correct.