Conditional lp contraction

For $p>1$, [F1] with $P(\mathrm{\Omega })=1$ makes $X$ integrable; for $p=1$ it is integrable by assumption. Put $U=E[X\mid \mathcal{G}]$. For finite $p$, [F2] supplies the convex Borel function and $E\mathrm{\mid }X{\mathrm{\mid }}^p<\mathrm{\infty }$ supplies its integrability. Jensen yields $\mathrm{\mid }U{\mathrm{\mid }}^p\le E[\mathrm{\mid }X{\mathrm{\mid }}^p\mid \mathcal{G}]$. Taking expectations using [F4] gives $E\mathrm{\mid }U{\mathrm{\mid }}^p\le E\mathrm{\mid }X{\mathrm{\mid }}^p$ and hence the norm inequality by taking the increasing positive pth root. At $p=1$ this is also the modulus estimate of [F4].

If $p=\mathrm{\infty }$, let $M=\mathrm{\parallel }X{\mathrm{\parallel }}_{\mathrm{\infty }}$. The inequalities $\mathrm{\mid }X\mathrm{\mid }\le M+1\mathrm{/}n$ for all positive integers hold outside a common null set; their limit gives $\mathrm{\mid }X\mathrm{\mid }\le M$ almost surely. Conditional order and constants imply $-M\le U\le M$ almost surely. Thus $\mathrm{\parallel }U{\mathrm{\parallel }}_{\mathrm{\infty }}\le M$. Equality of input representatives preserves the conditional class, so the maps are well defined on [F5]; linearity is [F4].