While the video camera setup would provide positive proof, it's probably not worth the time and effort.

Just get some printer's ink and liberally apply it to about 15 one dollar bills. Put all the bills in the wallet. Leave the wallet in the normal place. When the thief goes through it, there will be a telltale residue on his hands.

If he is quite smart, however, he may find some acetone to remove it before you can confront him.

There may be something better than printer's ink, such as the dye markers that are used to catch bank robbers, however, it might not be so easy to obtain it.