Do I have to pay a reward I offered for a lost item if I think the person who returned it stole it?
If you offer a reward to the public in return for information regarding the return of an object, and someone subsequently provides you with what you have requested, you are legally obligated to follow through with the reward offer. While you can certainly attempt not to pay it, the person in question has every right to take you to court in order to collect on the reward offer.
The Rules for Reward Offer and Acceptance
Stating that you are offering a reward for something is generally considered a "firm offer," and is basically the same as a contract. While the reward offer is not an offer that's specific to any one individual, it's being held out to the public in general and essentially states that anyone who completes the requested activity (i.e. return of jewelry, or producing information that leads to the return of the jewelry) is completing their end of the bargain, and you must also complete yours by paying the reward offer.
In situations regarding, for example, police reward offers or large companies offering rewards, it's possible for there to be paperwork created that states various loopholes and exemptions, which can sometimes void the requirement of paying the reward offer even if it appears the terms have been met. In the basic scenario, however, if you post a reward offer notice, it is a binding contract and any loopholes must be mentioned on the notice itself. Should someone fulfill what you are requesting, you must follow through with your end of the deal and pay the reward offer.
The rules for contracts can be complex, and there are a wide variety of different legal rules regarding when exactly something is considered a reward offer and when that offer is accepted and thus becomes binding. To get help understanding the rules of reward offers, and to make sure you don't accidentally find yourself owing a duty you don't want to owe or facing a breach of contract suit, you should always consult with a lawyer for guidance and advice.