What is predual space to Radon measures with finite moment? Define for any $p\in\mathbb{N}$ space $\mathcal{M}^p(\mathbb{R}_+)$ as measures with finite $p -$moment, i.e. such Radon measures $\mu$ that $\int_0^\infty x^p d\mu(x) <C_{p,\mu}$.


*

*What is the predual space of $\mathcal{M}^p(\mathbb{R}_+)$? 

*What is dual of $C^p(\mathbb{R}_+) -$ space of all continuous functions $f$ such that $\sup_{x\in\mathbb{R}_+}|\frac{f}{x^p}|< C_{p,f}$ ?

*Obviously I would like to have rigorous argument why I can intrgrate one agaist the other, although the previous two questions are important to me as they give me much more insight.

*Any comprehensive reference to similar spaces, their properties, duals and so on is much appreciated.


Best regards and thank you for your answer.
 A: [EDIT] What I said the other day was not quite right. Mostly right, the problem is that I referred to some things as complex measures that simply are not complex measures. (If we're just considering the real case then substitute "real measure" for "complex measure" everywhere.)
It's true that $\mathcal M^p$ is not the dual of anything, since it's not even a vector space (the OP assures us that he does intend for $\mathcal M^p$ to consist of just positive measures.)
And it's true that I doubt there exists a useful description of the dual of $C^p$, this being equivalent to characterizing the dual of the space of bounded continuous functions.
Now if we say $C^p_0$ is the space of continuous functions $f$ such that if $g(x)=f(x)/x^p$ then $g$ is bounded and $g(x)\to0$ as $x\to0$ and as $x\to\infty$ then we can say something about the dual of $C^p_0$. The map $f\mapsto g$ is an isometry onto $C_0$, and hence:
If $\lambda$ is an element of the dual of $C^p_0$ there exists a unique complex Radon measure $\mu$ such that $$\lambda f=\int_0^\infty f(t)/t^p\,d\mu(t).$$
That's the dual. The error was saying that we have $\lambda f=\int f\,d\nu$, where $\nu$ is the complex measure with $d\nu(t)=t^{-p}\,d\mu(t)$. For a general complex measure $\mu$ there is no such complex measure $\nu$, unless $\int t^{-p}\,d|\mu|<\infty$. (And if we restrict to such $\mu$ we don't get the whole dual.)
One could say that a general element of the dual is given by $$\lambda f=\int f\,d\nu,$$where $\nu$ is "something" such that by definition $$\int f\,d\nu=\int f(t)t^{-p}\,d\mu(t)$$for some complex measure $\mu$. But now we need to note that $\nu$ is not a complex measure and $\int f\,d\nu$ is not a Lebesgue integral.
This sort of thing happens, and this sort of integral notation for things that are not quite integrals does get used. For example people say that the dual of $H^1$ is BMO, and write the pairing as an integral, even though $fg$ is not integrable for $f\in H^1$ and $g\in BMO$. The integral is interpreted as a limit.
In this spirit one could say that although $\nu$, whatever it is, is not a complex measure, if $K$ is a compact subset of $(0,\infty)$ there is an actual complex measure $\nu_K$ which is equal to the restriction of $\nu$ to $K$. Which is to say that if $f$ has compact support then one can regard $\int f\,d\nu$ as an actual integral; now any function in $C^p_0$ can be approximated in norm by functions with compact support.
[Previous not-quite-right version]

The space $\mathcal M^p$ that you define cannot be the dual of anything since it's not a vector space. To make the question sensible you should define $\mathcal M^p$ to be the space of complex Radon measures such that $$\int_0^\infty x^p\,d|\mu|<\infty.$$(Possibly you meant "complex measure" when you said "measure"; that's a fairly standard usage in this context. But if so note that you still need to change the definition to refer to $|\mu|$ instead of $\mu$.)
The revised $\mathcal M^p$ is the dual of the space $C^p_0$, which I define to be the space  of continuous functions on $\Bbb R^+$ such that if $g(x)=f(x)/x^p$ then $g$ is bounded and $g(x)\to0$ as $x\to0$ and as $x\to\infty$.
This is easy to see. The space $C^0_0$ is the same as what's commonly known as $C_0$, and the Riesz Representation Theorem shows that the dual of $C_0$ is $\mathcal M^0$, the space of complex Radon measures. Now there is an isometric isomorphism $T:C_0\to C_0^p$ defined by $Tf(x)=x^pf(x)$ and an isometric isomorphism $S:\mathcal M^0\to\mathcal M^p$ defined by $S\mu=
\nu$, where $d\nu=x^{-p}\,d\mu$. These maps satisfy $$\int Tfd(S\mu)=\int f\,d\mu,$$so the result for $p=0$ implies the result in general.
I doubt that there exists a simple description of the dual of the space $C^p$. As above, this is equivalent to describing the dual of $C^0$, which is the space of bounded continuous functions. What the heck is that? (One could say that the dual is the space of Radon measures on the maximal ideal space, but I doubt that that's the sort of thing that's going to help.)
