# Open $\sigma$-compact sets with finite measure

Let $X$ be locally compact Hausdorff space and let $\mu$ be positive Borel measure, finite on compacts, outer regular with respect to open subsets, for each Borel set, and inner regular with respect to compact subsets, for each open set and for each Borel with finite measure. Is it true that for every compact $F$ there exists an open $\sigma$-compact $G$ such $F\subset G$ and $G$ has finite measure.

Thanks.

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