I have couple of questions about lemmas which lead to the proof of Szemeredi's Graph Regularity Lemma. I am using the book of Y.Zhao "Graph Theory and Additive Combinatorics". First of all I need to provide some basic definitions.
Definition 2.1.10 (Energy)
Let $G$ be an $n$-vertex graph. Let $U,W\subseteq V(G)$. Define $$q(U,W):=\frac{|U||W|}{n^2}d(U,W)^2.$$ For partitions $\mathcal{P}_{U}=\{U_1,\dots,U_k\}$ of $U$ and $\mathcal{P_W}=\{W_1,\dots,W_l\}$ of $W$, define $$q(\mathcal{P}_U,\mathcal{P}_W):=\sum \limits_{i=1}^{k}\sum_{j=1}^{l}q(U_i,W_j).$$ Finally, for a partition $\mathcal{P}=\{V_1,\dots,V_k\}$ of $V(G)$, define its energy to be $$q(\mathcal{P}):=q(\mathcal{P},\mathcal{P})=\sum \limits_{i,j=1}^{k}q(V_i,V_j)=\sum \limits_{i,j=1}^{k}\frac{|V_i||V_j|}{n^2}d(V_i,V_j)^2.$$
Lemma 2.1.11 (Energy never decreases under refinement) Let $G$ be a graph, $U,W\subseteq V(G)$, $\mathcal{P}_U$ a partition of $U$, and $\mathcal{P}_W$ a partition of $W$. Then $q(\mathcal{P}_U,\mathcal{P}_W)\geq q(U,W).$
Proof. Let $n=\nu(G)$. Let $\mathcal{P}_U=\{U_1,\dots,U_k\}$ and $\mathcal{P}_W=\{W_1,\dots,W_l\}$. Choose $x\in U$ and $y\in W$ uniformly and independently at random. Let $U_i$ be the part of $\mathcal{P}_U$ that contains $x$ and $W_j$ be the part of $\mathcal{P}_W$ that contains $y$. Define the random variable $Z:=d(U_i,W_j)$. We have $$\mathbb{E}[Z]=\sum_{i=1}^{k}\sum_{j=1}^{l}\frac{|U_i||W_j|}{|U||W|}d(U_i,W_j)=d(U,W)=\sqrt{\frac{n^2}{|U||W|}q(U,W)}.$$ We have $$\mathbb{E}[Z^2]=\sum_{i=1}^{k}\sum_{j=1}^{l}\frac{|U_i||W_j|}{|U||W|}d(U_i,W_j)^2=\frac{n^2}{|U||W|}q(\mathcal{P}_U,\mathcal{P}_W).$$ By convexity, $\mathbb{E}[Z^2]\geq \mathbb{E}[Z]^2$, which implies $q(\mathcal{P}_U,\mathcal{P}_W)\geq q(U,W)$.
Important remark: I want to emphasize that actually $\mathbb{E}[Z]\geq d(U,W)$. Please see my previous post. Anyway it does not change the proof. However, I guess it affects the proof of Lemma 2.1.13. See the details below.
Lemma 2.1.12 (Energy never decreases under refinement) Given two vertex partitions $\mathcal{P}$ and $\mathcal{P}'$ of some graph, if $\mathcal{P}'$ refines $\mathcal{P}$, then $q(\mathcal{P})\leq q(\mathcal{P}').$
Proof. Follows easily from the previous lemma (or see the book).
Lemma 2.1.13 (Energy boost for an irregular pair) Let $G$ be an $n$-vertex graph. If $(U,W)$ is not $\varepsilon$-regular, as witnessed by $A\subseteq U$ and $B\subseteq W$, then $$q(\{A,U\setminus A\},\{B,W\setminus B\})>q(U,W)+\varepsilon^4 \frac{|U||W|}{n^2}.$$
Proof.
Define $Z$ as in the proof of Lemma 2.1.11 for $\mathcal{P}_U=\{A,U\setminus A\}$ and $\mathcal{P}_W=\{B,W\setminus B\}$. Then $$\text{Var}(Z)=\mathbb{E}[Z^2]-\mathbb{E}[Z]^2=\frac{n^2}{|U||W|}(q(\mathcal{P}_U,\mathcal{P}_W)-q(U,W)).$$ We have $Z=d(A,B)$ with probability $\geq |A||B|/(|U||W|)$ (corresponding to the event $x\in A$ and $y\in B$). So $$\text{Var}(Z)=\mathbb{E}[(Z-\mathbb{E}[Z])^2]\geq \frac{|A||B|}{|U||W|}(d(A,B)-d(U,W))^2>\varepsilon \cdot \varepsilon \cdot \varepsilon^2.$$ Putting the inequalities together gives the claim.
But if we take into account what I wrote in the Important remark this proof does not work out. Indeed, we'll get that $\text{Var}(Z)\leq \frac{n^2}{|U||W|}(q(\mathcal{P}_U,\mathcal{P}_W)-q(U,W)).$ But the lower bound for $\text{Var}(Z)$ which I wrote above is not true anymore.
EDIT. My remark tells us that $$\text{Var}(Z)=\mathbb{E}[Z^2]-\mathbb{E}[Z]^2\leq \frac{n^2}{|U||W|}\left(q(\mathcal{P}_U,\mathcal{P}_W)-q(U,W) \right).$$ On the other hand, $$\text{Var}(Z)=\mathbb{E}[\underbrace{(Z-\mathbb{E}[Z])^2}_{\xi}]=\sum_{\omega\in \Omega}\xi(\omega)\mathbb{P}(\omega)\geq \sum_{\omega\in A\times B}\xi(\omega)\mathbb{P}(\omega).$$ We notice that here $\Omega:=U\times W$ is the probability space and $\forall \omega \in \Omega\quad \mathbb{P}[\omega]=\frac{1}{|U||W|}$. We notice that for each $\omega\in A\times B$, we have $\xi(\omega)=(Z(\omega)-\mathbb{E}[Z])^2=(d(A,B)-\mathbb{E}[Z])^2$. All this implies that $$\text{Var}(Z)\geq (d(A,B)-\mathbb{E}[Z])^2\frac{|A||B|}{|U||W|}.$$ From Lemma 2.1.11 we know that $\mathbb{E}[Z]\geq d(U,W)$ and we see that it does not imply what we want.
In Yufei Zhao's book, the quantity $e_G(X, Y)$ is defined as follows (Definition 2.1.1):
$$ e_G(X, Y) := |\{(x,y)\in X\times Y : xy \in E(G)\}|. $$ This means if $a,b \in X\cap Y$ and $ab \in E(G)$ then the edge $ab$ it is counted twice in $e_G(X,Y)$. With this convention you will have $\mathbb{E}[Z] = d(U,W)$.
Clarification: You are correct that $e_G$ is not the number of edges. It is a little bit different but if you take the above display as definition of $e_G$ then the proof goes through as it is. Again, it is NOT the number of edges, and that sentence is an oversight in Yufei's book I believe. Let me know if this clarifies your doubts.