In this paper we start from previous results obtained in  on the abstract space of Daniell-Loomis integrable functionsL, which is constructed like to the Daniell extension process, but without continuity assumptions on the elementary integral.
The localized integral is used to prove thatL consists of those functions whose local upper and lower integrals are equal and finite, or thatL is closed with respect to improper integration.
Our results are also holded in integration with respect to finitely additive measures.
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