Let $(X,\mathcal{F},\mu)$ be a $\sigma$-finite measure space. Prove that if $I: L^p(\mu) \to \mathbb{R}$ is a continuous linear functional then there exists $I^+$, $I^-$ continuous positive linear functionals such that $I=I^+ + I^-$.
I know the proof for the case that $I$ is a bounded linear functional. We define $I^+(f):=\sup\{I(g) : g \in L^p, 0\leq g \leq f\}$ and $I^-=I^+ - I$ and prove the statement.
But in this case I don't know what to do, and I can't link the proof for the bounded case because, of course, $I^+$ is defined with a supremum, which just make sense if we assume boundness of $I$ ... and I don't think that's a way to fix it.
My attempt was based on: try to define a continuous $I^+$ for simple functions and then use the fact that given $f\in L^p$ exist a sequence of simple functions $\varphi_n \to f$ and use continuity of $I^+$ to give $I^+ (f)$, but I can't do this in a useful way .
Anyone can help me?
I am assuming that $1<p<\infty$.By a well known result there exists $g \in L^{q}$ ($q=\frac p {p-1}$) such that $I(f)=\int fg$ for all $f \in L^{p}$ and all you have to do is define $I^{\pm}(f)=\int fg^{\pm}$. The same idea works for $p=1$ also.