Question

Example of outermeasure which is not continous from above

Answer 1

was trying to find an example of an outer Measure which is not continuous from below. These are the definitions I use

An outer measure on X is a function μ∗:P(X)→[0,∞] if it fulfills

μ∗(∅)=0

μ∗(⋃∞j=1Aj)≤∑∞j=1Aj

And an outer measure is continuous from below when for the sequence (Aj)j∈N with Aj⊂Aj+1 for alle j the equality

μ∗(⋃j=1∞Aj)=limj→∞μ∗(Aj)

Some results which might be helpful

All measures are continuous from below

All metric outer measures are continuous from below

So I search for an outer measure which isn't continuous from below.