This word translated "measure" is the same word from which we get our English word "canon." (Not "cannon" but "canon.") That word means "standard, limit, or rule." We talk about the canon of scripture—that is, what is standard for scripture.
What Paul is arguing (if I can put it this way) is that he has not taken on more than he should when he asserts his authority over these people—as their leader. And his argument is "which God appointed us." God gave him, or allotted to him, this authority over the Corinthians—and the Corinthian church. Another way to put it is that God assigned him this area in which to work.
Paul's broader assignment was to be apostle to the Gentiles (even as Peter was apostle to the Israelitish people). In one sense, because of that, he was to consider that wherever Gentiles lived was his area of work. Corinth was a Gentile city. Paul went there and preached the gospel, in an area assigned to him by God. A church was raised up [there] by God (as the result, though, of the preaching by the apostle Paul). And those people, raised up by God in the city of Corinth, were then handed over to the apostle Paul to further refine their understanding in the truth of God—in more particular and specific areas of the knowledge of God.
But it is clear from the context that there were others in the city of Corinth who were there preaching. And it is clear from the letter that they were not there to help Paul, and they were not there by his assignment of them. Rather, they were there to displace him. And these were people who were ostensibly "Christian" ministers.
I want you to see this because of the apostle's argument. He did have authority from God over that area of the world. He was very much aware of his area of responsibility, and he was also very much aware of other people's labors. And he was concerned that he live within the limits of his authority.
