Let we suppose to have a HV Tube or CRT . The filament is connected to the secondary of the transformer used to supply the filament.
At time $t=0$ the wires and cathode are neutrally charged. Then when we switch the power on, the filament begins to heat and free electrons are being evaporated from the cathode. Now I would come to conclusion that if the electron has left the metal there should remain a hole and the cathode would begin to be more and more positively charged, but this is not happening.
Are those electrons replaced? If yes, then from where?
Or if they aren't emitted at all, are they generated?
It might be clear from the picture that, electrons emmited by cathode never strikes the anode. The anode is used for accelerating the electron, not to recombine the electron.
Further question is: Where the electrons in anode come from? For accelerating the electron beam, energy is used proportional to the kinetic energy of electrons, but how the current flows if the electons never strike the anode?