We　study　theoretically　the　process　of　a　weak　measurement　of　charge　states　in　a　double　quantum　dot　performed　by　coupling　to　a　quantum　point　contact.　We　focus　on　the　interplay　of　the　measurement　process　with　phonon-induced　dephasing　of　delocalized　superpositions　of　electron　states　and　compare　the　time　scales　relevant　for　the　dynamics　of　the　open　system　and　for　the　measurement　process:　the　decoherence　rate,　the　localization　rate　of　the　state　conditioned　on　the　measurement　outcome　and　the　time　scale　on　which　information　on　the　system　state　is　gained　in　the　process　of　the　measurement.