Measure finite topology on the ring of measurable functions

Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">M</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="script">A</m:mi> <m:mo>,</m:mo> <m:mi>μ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> M(X,A,μ ) be the ring of all real-valued measurable functions constructed over a measure space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="script">A</m:mi> <m:mo>,</m:mo> <m:mi>μ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (X,A,μ ) . A topology on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">M</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="script">A</m:mi> <m:mo>,</m:mo> <m:mi>μ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> M(X,A,μ ) , called the F μ -topology, is introduced. It is realized that the component, the quasicomponent and the path component in this F μ -topology are identical. It turns out that the F μ -topology on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">M</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="script">A</m:mi> <m:mo>,</m:mo> <m:mi>μ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> M(X,A,μ ) becomes connected if and only if μ is an atomic measure of a special type. It is also proved that the F μ -topology is first countable when and only when μ is a hemifinite measure. Finally, it is shown that the second countability of the F μ -topology is equivalent to the hemifiniteness of the measure μ together with the countable chain condition of the F μ -topology.