First recall the definition of Borel regular finite measure $\mu$ on $\mathcal{P}([0,1])$
- Every Borel set is $\mu$ measurable.
- For every subset $A\subset [0,1]$, there exists a Borel set $B$ such that $A\subset B$ and $$\mu(A) = \mu(B).$$ Measures only satisfy condition $(1)$ are called Borel measures, and if $(2)$ is satisfied as well are called Borel regular measures .
My question is how do we intuitively understand the "regularity" here?
I think it has to do with the idea of continuity.
Look at the space of bounded measurable functions $B([0,1])$ with the uniform norm, its continuous dual space is the space of finite additive finite valued Borel measures.
If we look at the space of bounded continuous functions $C_b([0,1])$ with the uniform norm, its continuous dual space is the space of finite additive finite valued Borel regular measures.
How did continuity play the roll here to force the measures to be regular? And is there any examples where a Borel non-regular measure is clearly a continuous linear functional on $B([0,1])$ but not on $C_b([0,1])$?
Edit: And reading from wiki, I just want to add:
without the assumption of regularity in general, the uniqueness will fail. That is two Borel non-regular measures can act as two different continous linear functionals on bounded measurable functions. But when restricted to the continuous functions, their integrations give the same continuous linear functional.
Is there any good examples where we can see this?
I don't quite understand the example given in wiki
Without the condition of regularity the Borel measure need not be unique. For example, let X be the set of ordinals at most equal to the first uncountable ordinal Ω, with the topology generated by "open intervals". The linear functional taking a continuous function to its value at Ω corresponds to the regular Borel measure with a point mass at Ω. However it also corresponds to the (non-regular) Borel measure that assigns measure 1 to any measurable subset of the space of ordinals less than Ω that is closed and unbounded, and assigns measure 0 to other measurable subsets.