bat voltage should be min 13.5 to14.5 when under full electrical load,ielights,screen heater ,heater blower etc.if so then probable alternator is ok.battery also has cca (cold crank amperage)capacity.if this is low it will give symptons u indicate.it is entirly possible to have the low reading voltage u show and run the car until u have a situation that needs the cca reserve u dont have and engine fails to start.the only test that would confirm is a crank amperage test.alternitavly hooking up a jump start will definatly tell u ,if jump works ,new battery is needed.the fault is in the battery internals and is called plate sulfation,no cure.
caine, June 2008