simulation and measurement results are different. What should I do?

I simulated antenna on 5.9 GHz but when I measure it shows me that antenna works at 5.93GHz. What should I do?

It's normal.. 30 MHz shift is acceptable. There may be different reasons but simulators show only "prediction" in a acceptable accuracy.
If you tweak ( fine tuning on dimensions for instance ) you can make them consistent.

Yes it's just fine. in 2D/3D modeling & simulation you may specify the dielectric surrounding the antenna (note that air and vacuum don't necessarily have the same dielectic constant).