How do I eliminate the black border on a georeferenced map layer?

"Respect" is the correct term here. Respect has only one use in Saints Row 3, and that's to level up and unlock new upgrades. Generally then you buy the upgrades themselves with cash.

The highest Respect level is 50, and you can earn Respect in a million different ways - any time you see a little circle in the upper left with points counting upwards, you're earning Respect.

If you're Respect level 50, you get nothing from further Respect. Chances are by the time you finished the game you hit the cap, but you can still gain more easily through playing side missions or, really, doing whatever.

Specific choices have different outcomes when it comes to Respect or cash bonuses. For instance, at the end of act 1, your choice gives you 10% increase to either cash or Respect earned from that point onward.

Thus, the difference in the choices amounts to "not much" - if you picked Respect instead of cash, you'd hit the level cap faster, and earn cash a little slower.

The answer to the question in the title is: No, even on finite measured spaces.

For an example, consider $X=(0,1)$ endowed with the Lebesgue measure, and $f_n=2^n\cdot\mathbf 1_{(0,1/n)}$.