Rolf,
Turning a thin walled tube on a lathe can be a serious challenge. Just try seamless tubing, usually found at any good hardware store, hardware supply company, or automobile parts store. Usually less than $10 US for a 3 foot piece. This stuff will have an exact O.D. measurement, either English inches or metric mm. The I.D. will vary depending on the wall thickness you select. In the example above the tubing was 0.500 inch O.D. by 0.049 inch wall giving a bore of 0.402 inch. If you wish to rifle the tube bore after installation of the tube, I recommend that you do not use annealed tubing, use hard tubing as it is better behaved for cutting the rifle grooves.
Jim