On the Dutch website Dutchcowboys Arvid Bux spotted a photograph of a presentation from Google with information about measuring Google+ Pages. The photograph shows that actually quite logical Google Analytics will be used for measuring activity on Google+ Pages.
I was able to find the original presentation (Google Confidential) from Google with some more information about connecting Analytics and a Google+ Page.
To measure a page you only have to make a new profile in Analytics and then paste the unique tracking ID in the settings from the Google+ page, this is the same way you can currently measure Picasa Web Albums. Also there is a button to make a new Analytics profile directly from within the page. By measuring a page this way the statistics are separately from a existing profile so you can’t combine it with a existing profile from your own company website.
After adding the code you can, according to the slides, see the total number of visits, pageviews, average time on site, bounce rate, etc. just like you are used to when measuring websites. At the top of the page there is a new button (next to the circles button) to reach the statistics.
At least Audi USA is currently testing this connection but I guess other big companies are testing this as well. No word on when the Google Analytics connection is actually coming available for all Google+ Pages, I hope it will not take to long :-).
The slides from the original presentation below, click to see a bigger version.