An unbounded predictable transform may lose integrability

Let $\mathrm{\Omega }=\{(j,e):j\ge 1,e\in \{-1,1\}\}$ with its full power-set sigma-algebra and masses $P\{(j,e)\}=2^{-j-1}$. The weighted Dirac sum is a measure by [F1]–[F2]. Its first m two-point blocks have total mass ${\sum }_{j=1}^m{2}^{-j}=1-2^{-m}$, by multiplying the finite sum by two and subtracting; since $2^m\ge m+1$, its limit is one. Hence $P(\mathrm{\Omega })=1$. Let ${\mathcal{F}}_0$ be all unions of the blocks $B_j=\{(j,-1),(j,1)\}$ and ${\mathcal{F}}_n$ the full power set for $n\ge 1$. Complements and countable unions preserve block unions, so this is a filtration.

Set $M_0=0$, $M_n(j,e)=e$ for $n\ge 1$. These variables are adapted and bounded by one. For any $A\in {\mathcal{F}}_0$, its positive-sign and negative-sign portions have equal probability: each equals ${\sum }_{j:B_j\subseteq A}{2}^{-j-1}$. Thus ${\int }_{A}edP=0$, a finite subtraction, so the zero function meets all CE event identities. Consequently $E[M_1\mid {\mathcal{F}}_0]=0=M_0$. At later times $M_{n+1}=M_n=e$ is already known and integrable, so it conditions to itself. Thus $M$ is a bounded martingale Martingale submartingale and supermartingale.

Define $H_1(j,e)=2^j$ and $H_k=0$ for $k\ge 2$. Every value is finite, $H_1$ is constant on each block and hence ${\mathcal{F}}_0$-measurable, and later H values are known constants. Thus $H$ is predictable Predictable discrete time process and unbounded. The algebraic gain at every positive time is $G_n(j,e)=2^{j}e$. For $f_m=2^j{\mathbf{1}}_{\{j\le m\}}$, the finite simple integral is $E{f}_m={\sum }_{j=1}^m{\sum }_{e=\pm 1}{2}^j{2}^{-j-1}=m$. To justify the limiting step locally, augment every finite disjoint simple display by its complement with coefficient $0$; intersections of two augmented displays partition $\mathrm{\Omega }$ and carry equal coefficients, so finite additivity and $0\cdot (+\mathrm{\infty })=0$ prove representation independence. Common refinements give monotonicity. For any $0\le u_r\uparrow u$, a simple $s\le u$, and $0<c<1$, the sets $A_r=\{u_r\ge cs\}$ increase to $\mathrm{\Omega }$, including on the zero level of $s$, and continuity from below for the finite-sum measure $A\mapsto {\int }_{A}s$ gives $c\int s\le {\sup }_r\int u_r$. Let $c\uparrow 1$ and take the supremum over $s$ to obtain MCT. Applying this to $f_m\uparrow \mathrm{\mid }{G}_1\mathrm{\mid }$ gives $E\mathrm{\mid }{G}_1\mathrm{\mid }=\mathrm{\infty }$. Hence the product at time one is not integrable and violates [F7], and the gains cannot be a martingale. No signed conditional expectation of $G_1$ is formed. AC is inherited only from the CE notation used for the bounded M; all masses and factors are explicit.