### Control functions and total boundedness in the space ${L}_{0}$.

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The $\sigma $-finiteness of a variational measure, generated by a real valued function, is proved whenever it is $\sigma $-finite on all Borel sets that are negligible with respect to a $\sigma $-finite variational measure generated by a continuous function.

A descriptive characterization of a Riemann type integral, defined by BV partition of unity, is given and the result is used to prove a version of the controlled convergence theorem.

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