By the Outer Space Treaty - you can't own the Moon - and nor can any country.
However what you can own are habitats. If you build a colony on the Moon - you will own the habitats that you build. So long as you occupy them at least.
Also you probably have some right to safety region around your habitat e.g. if someone tries to land rockets just next to your habitat and endangers it.
This may be developed into some kind of property rights - but the basic principle of the OST is - that nobody can own territory on the Moon - e.g. for a military base or whatever.
Or anywhere else in space for that matter. And I think most would agree that is a good principle to hold to. At least when you think in terms of an alternative of a solar system divided up between various military groups who own parts of it and defend it against incomers.
And just about all countries have signed the treaty - certainly all ones with space faring ambitions (including N. Korea) and most of the ones that have no current plans at all in space as well, even tiny countries.
So - no - no law to prevent you building your colony.
However, "assets" are another matter. If you try to lay claim to resources on the Moon - say the polar ice for example maybe one of the most valuable deposits there - then you have no legal basis for that claim at present.
So your claim would be disputed - nobody would recognize it. The thing is - as no country or entity owns any solar system object or region outside of the Earth - there is no-one with the authority to grant you that claim. And indeed in the OST then is clear that all resources in space are to be regarded as for the benefit of all humanity - though what that means in practice is yet to be worked out.
Also - though it hasn't happened yet - the idea has been discussed of special "parks" on the Moon. That would probably include all the historical landing sites and the first footprints on the Moon - that you can't just go and build on them or take them back to Earth and sell them or whatever.
Could just be an informal agreement - or might become law in some sense.
And - also - sensitive science areas - e.g. maybe the polar ice regions could be sensitive to contamination or whatever - then other countries - if that was so - then under the OST then they can say that you are engaging in activities that are harmful to their own scientific investigation of the Moon.
So - that's mainly been used to protect planets from contamination by Earth life. But - it might apply more generally than that.
(Note - I'm not a lawyer - this is just a lay person's view of the legal situation and the OST from what I've read).