Constructing a premeasure on half open intervals

[c4rpe]

I want to construct a premeasure on half open intervals in the set of extended real numbers and showing
the steps of construction clearly.
I know what is premeasure and half open interval of course, but I have problems with showing my steps clearly. Please help

 

[c4rpe]

any idea?

 

[micromass]

What do you mean with a premeasure?