On the density topology with respect to an extension of Lebesgue measure (Q1374572)

The author studies the density topology generated by complete extensions of Lebesgue measure. He proves that for any complete extension \(\mu\) of Lebesgue measure the corresponding \(\mu\)-density topology is the Hashimoto topology of the form \(\tau^*_\mu =\{X\subset \mathbb R: X=Y\setminus Z,\^^MY\in \tau_d,\;\mu (Z)=0\}\), where \(\tau_d\) is the density topology derived from the Lebesgue measure. As a consequence of the theorems proved in the article the author obtained the following: For any complete extension \(\mu\) of Lebesgue measure, the family of the real functions continuous in the \(\mu\)-density topology coincides with the family of all approximately continuous functions.