We are starting an RSS feed, and have placed ad banners to the extended news text pages where RSS feeds link to. This gives some estimate on daily and monthly visitors who click from the feed to our website, but of course doesn't say how many people read just the feed itself.
You can use your log/analyitics software and see how many requests you are getting to your feed. That would be how many times it is being reqested by a news aggregator tool/service, search engine/robot, or a user. The downside to this is that it may be requested numerous times from the same user or IP which would skew the results.
That is the only measurable I know of right now. I would love to find out more!
I think everyone who has suggested RSS and has to report to a client has this problem. There are a lot smarter people than me so I went higher than me to people who really know and below is the answer I got from Bill Kearney at Syndic8.
When publishing a feed make sure the links you use are something you can track in your logs as having been used in the RSS feeds.
You can use FeedBurner for this as well. If you proxy your feed through FeedBurner, we'll give you a number of stats including the circulation of your feed and, optionally, we'll rewrite the link elements to give you detailed click-through statistics.
The one tricky thing with calculating the number of subscribers (that we've invested quite a bit of time in) is categorizing the different kinds of user-agents that hit your feed into bots, browsers, aggregators ("server-based reader"), and clients. Bots and browsers don't generally "count" as subscribers, while a single hit from an aggregator like Bloglines or My Yahoo may represent a number of readers.