Non-Commutative Integration

We will show that if $\text{sM}$ is a factor, then for any pair $\text{f},\text{p}\in \text{sMdsup}$ of normal positive linear functionals on $\text{sM}$, the inequality: $$\text{lrnorm}\text{f}\le \text{lrnorm}\text{p}$$ is equivalent to the fact that there exist a countable family $\text{lrbrace}\text{ffdi}:i\in I\subset \text{sMdsup}$ in $\text{sMdsup}$ and a family $\text{lrbrace}\text{udi}:i\in I\text{i}\text{sM}$ of partial isometries in \cM such that $$\text{f}=\text{sumd}i\in I\text{ffdi},\text{sumd}i\in I\text{udi}\text{ffdi}\text{udius}\le \text{p},\text{and}\text{udius}\text{udi}=s\text{lr}\text{ffdi},i\in I,$$ where $s(\omega), \omega\in\sMdsup$,meansthesupportprojectionof$\omega$\mathrm{.}Furthermore,if$\lrnorm{\f}=\lrnorm{\p}$, then the equality replaces the inequality in the second statement. In the case that $\text{sM}$ is not of type \threeonec the family of partial isometries can be replaced by a family of unitaries in \cMp One cannot expect to have this result in the usual integration thoery. To have a similar result, one needs to bring in some kind of non-commutativity. Let $\text{lrbrace}X,\mu$ be a $\text{sig}$-finite semifinite measure space and $G$ be an ergodic group of automorphisms of $\text{linflr}X,\mu$, then for a pair $f$ and $g$ of $\mu$-integrable positive functions on $X$, the inequality: $${\int }_{X}f(x)\text{txd}\mu (x)\le {\int }_{X}g(x)\text{txd}\mu (x)$$ is equivalent to the existence of a countable families $\text{lrbrace}\text{fdi}:i\in I\subset L^1(X,\mu )$ of positive integrable functions and $\text{lrbrace}\text{gdi}:i\in I$ in $G$ such that $$f=\text{sumd}i\in I\text{fdi}\text{and}\text{sumd}i\in I\text{gdi}\text{lr}\text{fdi}\le g,$$ where the summation and inequality are all taken in the oredered Banach space $L^1(X,\mu )$ and the action of $G$ on $\text{lonelr}X,\mu$ is defined through the duality between $\text{linflr}X,\mu$ and $\text{lonelr}X,\mu$, i.e., \lr{\g(f)}(x)&=f\lr{\g\inv x}\frac{\txd\mu\scirc \g\inv}{\txd\mu}(x), \quad f\in\lonelr{X, \mu}.