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I'm reading chapter 6 (measure theory) of Pedersen's book Analysis now and I'm a bit puzzled in his passage from Radon integrals to Radon charges. The book in 6.1.2 defines a Radon integral to be a linear complex valued positive functional (i.e which maps positive functions in positive real numbers) defined on the space $C_c(X)$ of continuous (complex valued) functions with compact support on a locally compact Hausdorff space $X$. In 6.5.5 the book gives three definitions of a Radon Charge:

  1. A Radon charge is a linear combination of Radon integrals.

  2. Let $\tau$ be the weak topology on $C_c(X)$ given by the family of semi norms $f\mapsto |\int f|$ where $\int$ is a Radon integral and $f\in C_c(X)$. A Radon charge is a continuous linear functional with respect to this topology.

  3. A Radon charge is a linear functional $\Phi:C_c(X)\to\mathbb{C}$ such that $$\sup\{|\Phi(g)|:g\in C_c(X),\, |g|\leq |f|\}$$ is finite for all $f\in C_c(X)$.

I can see the equivalence of 1 and 2. In theorem 6.5.6 it is proved that 3 holds for $\Phi$ iff $\Phi(f)=\int (f\cdot u)$ with $\int$ a Radon integral and $u: X\to \mathbb{C}$ a Borel function of modulus one. I don't see why this theorem sheds any light on the equivalence of 3 with 1 or 2 above. Can someone help me?

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If you have

$$\Phi(f) = \int f\cdot u,$$

write $u = g^+ - g^- + ih^+ - ih^-$ with non-negative Borel functions $g^+,g^-,h^+,h^-$. Let $\Phi_1(f) = \int fg^+$, $\Phi_2(f) = \int fg^-$, $\Phi_3(f) = \int fh^+$, $\Phi_4(f) = \int fh^-$. Then the $\Phi_i$ are Radon integrals, and

$$\Phi = \Phi_1 - \Phi_2 + i\Phi_3 - i\Phi_4$$

is a linear combination of Radon integrals.

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  • $\begingroup$ thanks! very neat and elementary…. $\endgroup$ – mathlearner Nov 17 '13 at 21:32

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