How can I tell if my television has a digital tuner in it?

You should be able to check the back of the TV. Unless it's a pretty new TV, you are going to be out of luck for an RV TV.

All televisions purchased after March 2007 have built-in digital tuners. If your TV is older than that, there's a good chance you don't have DTV.

If you have an analog TV, you will have to purchase a digital-to-analog set-top converter box to watch over-the-air digital programming. The converter will attach to your current TV.

Now that the Digital transition has official been made in mid-2009 (after an initial delay date) by the Federal Communications Commission and USA Government mandating requirements for all full power stations to cease broadcast on the Analog Side and broadcast a Digital Quality Signal.
NOW ON TO YOUR ANSWER !
The easiest way to determine digital compatibility of your RV set would be to turn the T.V. and dial around and see if you are able to tune (or see if not) you are receiving channels.
If you are not getting your local stations on the low end, such as channels 2,3,4,5,6,7,8,9,10- you definitely need a converter box to watch over-the-air digital programming.
You can get coupons to defray about 90% of the cost! Go to this site https://www.dtv2009.gov/ApplyCoupon.aspx