Can I measure current in antenna? What is the difference between current in closed AC circuit and current in antenna?
1) Yes of course you can measure antenna current. It just takes a suitable ammeter.
Whether this makes sense or not is just a matter of what is your application.
Antenna current is commonly measured in MW/SW high power transmitters, it's mostly done to monitor output power or as an input to automatic antenna tuner.
Nowadays a simple current transformer is invariably used while in the early radio era this was often a hot wire ammeter.
In other fields, like higher frequencies or lower power or in receiving applications it is much easier to monitor some other quantity instead of antenna current.
2) There's no difference between antenna current and closed circuit current it can simply prove hard or impossible to wire an ammeter somewhere in an antenna without disturbing its behaviour.
E.g. you can easily place a small ammeter halfway one of the brances of a 20m transmitting dipole and read (maybe with a binocular) less current than the feeding point.
But you cannot absolutely do anything on a 20mm cellphone GHz range antenna.
The antenna is a source of energy. You can "measure" it. Cellphones are required to tolerate strong "blockers" within 10 or 20MHz of the subscriber's channel, those blockers at levels of -10dBm. Given 0dBm is 0.632 volts PeakPeak across 50 ohms, the -10dBm is 1/sqrt(10) weaker at 0.2 volts, at 1Ghz or 2GHz or 5GHz. Spectrum Analyzers certainly handle these levels.
The normal cellphone power level, into the receiver, will be
+60dB for 1MHz channel bandwidth (TDMA, CDMA)
+5dB for SignalNoiseRatio acceptable to the Bit Error Correction
+5dB for input diplexor losses and Pi matching losses
total: -174 + 60 + 5 + 5 = -174 + 70 = -104dBm
With -100 dBm being 0.632vpp/100,000 or 6 microvolts peakPeak, your scopes cannot view the RF. But that tuned receiver the Spectrum Analyzer can measure, and more importantly display the frequencies used.