I tried to prove Riesz representation theorem for continuous linear functional defined on the continuous functions of compact support of a topological locally compact Hausdorff space in this way: 1) extending via Hahn-Banach theorem the continuous functional to all bounded Borel-measurable function in a norm preserving way; 2) defining a set function evaluating this extension on characteristic function of Borel-measurable sets and showing that it is actually a signed measure; 3) showing that this signed measure represents the functional via integration; 4) showing that this association is an isometry.
The only problem I have found is in the second step (it seems to me that if this step is true than the third and the fourth are easily proved). Here I found that it is simple to show finite additivity and even bounded variation of this map, but I don't know how to show sigma-additivity (I don't even know if this is possible to do... maybe it's false and the proof of the extension via Hahn-Banach doesn't work).
Does anyone know how to fix this gap?
Thanks