Jtdylktuy

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 $I^+(f):=sup{I(g) : g in L^p, 0leq g leq f}$ 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 $fin L^p$ 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.