pr.probability – “Cut norm” of conditional expectation has supremum on products of sets in sub-$sigma$-algebra, or not?

I am reading Lovasz’s book “Large networks and graph limits”, and encountered the exercise that the stepping operator for graphons is contractive under the cut norm:


Originally, the stepping operator from some partition $P:(0,1)=X_1cupcdotscup X_k$ acting on some graphon $W:(0,1)^2rightarrowmathbb{R}$, which is bounded, measurable and symmetric, is defined as below:

$$W_P(x,y)=frac{1}{lambda(X_i)lambda(X_j)}int_{X_itimes X_j}W(x,y)dxdy,forall (x,y)in X_itimes X_j,$$

where $lambda$ is the Lebesgue measure. And the cut norm of a graphon is defined as

$$||W||_square=mathop{sup}limits_{S,Tsubset (0,1)}|int_{Stimes T}W(x,y)dxdy|,$$

where $S,T$ are both measurable. I have figured out how to do this, starting by first prove that the supremum in the cut norm of $W_P$ is the same as the supremum on all $S,T$ generated by the partition. This is done by considering each small pieces of maximum $S,T$(merely measurable) in each partition parts $X_i$, moving them using measure-preserving maps to form intervals(this can be done since when we work with $W_P$, it’s constant on every small blocks), and discussing the boundaries.

Now I am thinking that, since the stepping operation can be viewed as taking conditional expectation given the $sigma$-algebra generated by $mathcal{P}$:


here I am using $mathcal{P}$ for the $sigma$-algebra generated by $P$. Does this general statement below also hold?

Fix some probability triple $(Omega,mathcal{F},mathbb{P})$, and a bounded, real-valued (symmetric)random variable $X$ on the product probability space $(Omega,mathcal{F},mathbb{P})otimes(Omega,mathcal{F},mathbb{P})$. Let $mathcal{G}$ be some sub-$sigma$-algebra. Do we have the following:
$$mathop{sup}limits_{S,Tinmathcal{F}}|mathbb{E}(mathbb{E}(Xmidmathcal{G}otimesmathcal{G})1_{Stimes T})|=mathop{sup}limits_{S’,T’inmathcal{G}}|mathbb{E}(mathbb{E}(Xmidmathcal{G}otimesmathcal{G})1_{S’times T’})|?$$

I tried to mimick the proof for graphons, which means $mathcal{G}$ is generated by finite number of sets. So at first I have

$$mathbb{E}(mathbb{E}(Xmidmathcal{G}otimesmathcal{G})1_{Stimes T})=mathbb{E}(Xmathbb{E}(1_{Stimes T}midmathcal{G}otimesmathcal{G})),$$

and $mathbb{E}(1_{Stimes T}midmathcal{G}otimesmathcal{G})=mathbb{E}(1_Smidmathcal{G}otimesmathcal{G})otimesmathbb{E}(1_{T}midmathcal{G}otimesmathcal{G})$, since they equal on cylinder sets in $mathcal{G}otimesmathcal{G}$. THen I got stuck, since I don’t know the structure of mathbb{E}(1_Smidmathcal{G}otimesmathcal{G}), unlike in finite sub-$sigma$-algebra case where I can discuss one-by-one in blocks.

Any help is desired.