Riesz representation theorem for $C_{0}([0,T])$

Question

I have a question about the Riesz representation theorem.

Let $X$ be a compact Hausdorff space and $C(X)$ the Banach space of all continuous functions on X with supremum norm. The Riesz representation theorem says that every bounded linear operator on $C(X)$ is realized by an integral with respect to an certain finite signed measure on $X$

I am trying to apply this theorem for the following Banach space.

$T>0$ : fix.

$C([0,T]):=\{w:[0,T]\to \mathbb{R}\,;\, w \,{\rm is\,conti.} \}$

$C_{0}([0,T]):=\{w \in C([0,T]) \,; \,w(0)=0 \}$

Then $C([0,T]),C_{0}([0,T])$ is Banach space with supremum norm $\|w\|=\sup_{0 \leq t \leq T}|w(t)|$

According to the above theorem, for any $A \in C([0,T])^{*}$, there exists an certain finite signed measure on $[0,T]$

Let $A \in C_{0}([0,T])^{*} $. Then can we deduce a finite signed measure corresponds to $A$ is a finite signed measure on $(0,T]$?

Answer 1

Hint Apply Riesz representation theorem for the operator $$Bf := A(f-f(0)), \qquad f \in C([0,T]).$$

Answer 2

If $\Phi \in C_{0}[0,T]^{\star}$, then, by the Hahn-Banach Theorem, there exists $\Psi \in C[0,T]^{\star}$ such that $\|\Phi\|=\|\Psi\|$ and $\Psi=\Phi$ on $C_{0}[0,T]$. That yields the existence of a finite signed Borel measure $\mu$ on $[0,T]$ such that $$ \Phi(f) = \int_{0}^{T}f\,d\mu. $$ The measure $\mu$ is unique among the finite Borel signed measures for which $\mu\{0\}=\mu\{T\}=0$. This uniqueness follows from fact that the characteristic functions of open intervals $(a,b)\subseteq (0,T)$ are pointwise limits of uniformly bounded functions in $C_{0}[0,T]$.