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During the rise and peak of Islam, a very structured and organized law system was established based according to the Quran and religion which had been highly depended upon. Due to this, many people did not commit crimes, steal, or lie, knowing that their Lord may punish them. Therefore, this all comes down to establish the very core of the matter: the caliph set laws and punishments which were all agreed upon by scholars, community elders, and respected Muslims.
The caliphs of that age had a system which they called Hokm Islami. This translates to Islamic rule without force. Later, the Ottomans established the forced rule for all, based on certain groups and religion types.
Abu Bakr set many limits on himself as other Caliphs had done. This included equality of wealth for all. There had been a treasury which they called "Bait Ul-Mal" , which the Caliph was not allowed to take money from, unless it was to help the community or poor folks. The Caliph also limited himself to water and basic food to live. He lived in a humble home as other men did.
Aside from money, the Caliph Abu Bakr also made sure that he did not order anything unless it was backed by the Quran, and agreed upon.
I hope this helped. If not, please comment below and I'd certainly be glad to elaborate and add to this description. Thanks! :)
