Well, the burden of proof, as in every criminal trial, is that the jury or the judge have to be convinced beyond a reasonable doubt that the accused has committed the crime.
The prosecutor basically has to prove two things:
a) that a crime has been committed
b) that the crime has been committed by the defendant
Now, how can a prosecutor prove this:
a) either by direct evidence, for example an eyewitness who saw it all happen
b) by circumstancial evidence, that is facts and circumstances from which the jury or judge may reason that the above elements were proven.
So, is it necessary to prove petty theft with a video tape? Not at all. An eyewitness can even be a much better proof than a video tape, because he can be questioned as to the circumstances of his observations.
If there is an eyewitness who says he saw the crime happening and who can identify the perpetrator it is up to the jury or the judge to believe him. If they do this might be all the prosecutor needs to get a conviction.