Let $G$ be a locally compact Hausdorff topological group, and denote by $B$ the σ-algebra generated by the open subsets of $G$, an element of $B$ is called a Borel set.
A (left) Haar measure $m$ on $G$ is a measure $m$ on $B$ which is:
outer regular on all Borel sets $B$
inner regular on all open sets $U$
finite on all compact sets $K$
invariant under left translations: $m(gB)=m(B)$ for all Borel sets $B$ and all $g$ in $G$
- nontrivial: $m(B)> 0$ for all non-empty open sets $B$.
Wikipedia's Haar measure article says
The left Haar measure satisfies the inner regularity condition for all σ-finite Borel sets, but may not be inner regular for all Borel sets.
Can we conclude that Haar measure $m$ is inner regular on all finite Borel sets $B$, just making use of the conditions above?
Can we conclude that Haar measure $m$ is inner regular on all σ-finite Borel sets if $m$ is inner regular on all finite Borel sets $B$?