What problem did Edwin Hubble try to solve?

Hubble is famous for observations that he made when he wasn't trying to solve any problem at all. He was measuring the red-shift of galaxies of different brightness (he presumed dim galaxies to be farther away). He was surprised to find that red-shifts increased with dimness. But this was pure serendipity--an accidental discovery. He wasn't trying to prove anything. And in fact he never did come right out and state the obvious conclusion--that the whole universe was rapidly expanding. Other scientists had to state the conclusion for him.

The 'Redshift' problem.

He noted that the further away an object on average,

the more it's doppler shift indicated its recession.