[Bubba Ho-Tep]

If they remained a part of the union why then did Grant occupy Paducah?

Because Gideon Pillow had invaded and occupied Columbus.

[JCBreckenridge]

Then Kentucky was not a union state at the start of the war. Neither was New Mexico Territory, Arizona Territory.

There were confederate supporters in Missouri and in Maryland as well that controlled parts of those states at the start of the war.

BTW - the legimate authority in Kentucky, Magoffin, condemned Grant for his invasion of Paducah.