> UPDATE: if money is lying around on the street you are not be allowed to keep it (the same as I should not be allowed to keep the "money lying around in the internet"), but you can claim for getting the legitimate finder's reward.
It's actually a bit more complicated than that. Property (chattels) can be lost, mislaid, or abandoned. The distinction between lost and mislaid is that you mislay something when you intentionally put it somewhere but forget to retrieve it. That doesn't really apply to currency in the street -- it's unlikely that someone would intentionally leave money in the street.
So we are dealing with lost property. At common law the rule was that the finder of a chattel in a public place had a superior title to anyone except the true owner, and that if such a possessor knew or could reasonably ascertain the owner's identity he had a duty to notify the owner. A breach of that duty could result in either tort or criminal liability or both. In the case of fungible currency in the street, I'd say there's a good argument that it is not reasonable to ascertain the true owner.
However this common law rule has been modified in most jurisdictions by statute. Illinois has a typical such statute:
(765 ILCS 1020/27-8)
Sec. 27. If any person or persons find any lost goods, money, bank notes, or other choses in action, of any description whatever, such person or persons shall inform the owner thereof, if known, and shall make restitution of the same, without any compensation whatever, except such compensation as shall be voluntarily given on the part of the owner. If the owner is unknown and if such property found is of the value of $100 or upwards, the finder or finders shall, within 5 days after such finding file in the circuit court of the county, an affidavit of the description thereof, the time and place when and where the same was found, that no alteration has been made in the appearance thereof since the finding of the same, that the owner thereof is unknown to the affiant and that the affiant has not secreted, withheld or disposed of any part thereof. The court shall enter an order stating the value of the property found as near as the court can ascertain. A certified copy of such order and the affidavit of the finder shall, within 10 days after the order was entered, be transmitted to the county clerk to be recorded in his estray book, and filed in the office of the county clerk. ...
Sec. 28. In all cases where such lost goods, money, bank notes or other choses in action shall not exceed the sum of $100 in value and the owner thereof is unknown, the finder shall advertise the same at the court house, and if the owner does not claim such money, goods, bank notes or other choses in action within 6 months from the time of such advertisement, the ownership of such property shall vest in the finder and the court shall enter an order to that effect.
If the value thereof exceeds the sum of $100, the county clerk, within 20 days after receiving the certified copy of the court's order shall cause a notice thereof to be published for 3 weeks successively in some public newspaper printed in this county and if the owner of such goods, money, bank notes, or other choses in action does not claim the same and pay the finder's charges and expenses within one year after the advertisement thereof as aforesaid, the ownership of such property shall vest in the finder and the court shall enter an order to that effect.
