BH independent FDR

For every $k\in \{1,\dots ,m\}$ and real $a$, $\{P_{(k)}\le a\}={\bigcup }_{J\subseteq \{1,\dots ,m\},\mathrm{\mid }J\mathrm{\mid }=k}{\bigcap }_{j\in J}\{P_j\le a\}$, since the $k$th smallest value is at most $a$ exactly when at least $k$ coordinates are at most $a$; the half-line is Borel by [F3], so [F2] makes this finite union of measurable intersections measurable, and the same formula works after deleting any coordinate.

Each $A_k=\{P_{(k)}\le kq\mathrm{/}m\}$ is measurable by step 1.1; its indicator and the simple function $k{\mathbf{1}}_{A_k}$ are measurable by [F2], and the finite maximum of these functions and $0$ is measurable because its sublevel sets are finite intersections. The same construction with $P_{-i,(k)}$ and thresholds $(k+1)q\mathrm{/}m$ shows each level set of $S_i$ lies in the sigma-algebra generated by the remaining coordinates.

If $R=0$, no $P_i$ is zero because then $P_{(1)}=0\le q\mathrm{/}m$, so no value is at most $Rq\mathrm{/}m=0$. If $0<R<m$, maximality gives $P_{(R)}\le Rq\mathrm{/}m$ and $P_{(R+1)}>(R+1)q\mathrm{/}m>Rq\mathrm{/}m$, so exactly the first $R$ ranks have values at most $Rq\mathrm{/}m$; if $R=m$, all values are at most $P_{(m)}\le q=mq\mathrm{/}m$. Thus the rejected indices are exactly $\{i:P_i\le Rq\mathrm{/}m\}$.

Fix $i$ and write $s=S_i$. If $s\ge 1$, then $P_{-i,(s)}\le (s+1)q\mathrm{/}m$, and if $s=0$ there are no other values to check. When $P_i\le (s+1)q\mathrm{/}m$, at least $s+1$ full-family values meet that threshold, so $R\ge s+1$. If $R\ge s+2$, the qualifying rank $R$ has $P_{(R)}\le Rq\mathrm{/}m$; deleting $P_i$ leaves at least $R-1$ other values at most $Rq\mathrm{/}m$, making $R-1\ge s+1$ qualify in the definition of $S_i$, a contradiction. Hence $R=s+1$, and step 3.1 shows that $i$ is rejected.

If $P_i>(s+1)q\mathrm{/}m$ but $i$ is rejected, step 3.1 gives $P_i\le Rq\mathrm{/}m$; write $r=R\ge 1$. At least $r-1$ other values are at most $rq\mathrm{/}m$, so $r-1$ qualifies in the definition of $S_i$ (with $0$ automatically qualifying when $r=1$). Thus $r-1\le s$, whence $P_i\le rq\mathrm{/}m\le (s+1)q\mathrm{/}m$, a contradiction; therefore $i$ is not rejected.

Step 4.1 shows that $P_i\le (s+1)q\mathrm{/}m$ implies $I_i=1$ and $R=s+1$, while step 4.2 shows that otherwise $I_i=0$; substituting these two cases gives the stated pointwise identity.

Fix $\theta$ and work on the probability space $(\mathcal{X},\mathcal{A},P_{\theta })$, regarding each $P_i:=p_i$ as a random element; [F3] makes these functions measurable. For $k\in \{1,\dots ,m\}$ and $a\in \mathbb{R}$, $\{P_{(k)}\le a\}={\bigcup }_{J\subseteq [m],\mathrm{\mid }J\mathrm{\mid }=k}{\bigcap }_{j\in J}\{P_j\le a\}$, so each rank-threshold event is measurable. The events $\{R=r\}$ are finite Boolean combinations of the rank-threshold events, so $R$ is measurable. If $0<R<m$, maximality in [F2] gives $P_{(R)}\le Rq\mathrm{/}m$ and $P_{(R+1)}>(R+1)q\mathrm{/}m>Rq\mathrm{/}m$, so the rejected hypotheses are exactly those with $P_i\le Rq\mathrm{/}m$. If $R=m$, all values are at most $q=Rq\mathrm{/}m$; if $R=0$, no value is zero because then rank $1$ would qualify. Hence the same characterization holds in every case. The rejection events are measurable because $\{P_i\le Rq\mathrm{/}m\}={\bigcup }_{r=0}^m(\{R=r\}\cap \{P_i\le rq\mathrm{/}m\})$, so the BH rule is a measurable map into the finite set $2^{[m]}$. Now fix $i\in I_0(\theta )$. [A1] and [F4] give independence; [F5] groups $P_i$ and the remaining $P_j$, while [F7] makes $S_i$ measurable with respect to the latter group. Thus $P_i$ and $S_i$ are independent random elements, and [F6] gives their product joint law.

For $r\in \{0,\dots ,m-1\}$ put $t_r=(r+1)q\mathrm{/}m\le q<1$. By [A1], $P_{\theta }(P_i\le t_r)=t_r$; independence from step 1.1 and the product law [F6] give $P_{\theta }(P_i\le t_r,S_i=r)=t_r{P}_{\theta }(S_i=r)$.

By [F7], partitioning on the finite-valued $S_i$ expresses $I_i\mathrm{/}(R\vee 1)$ as ${\sum }_{r=0}^{m-1}\mathbf{1}\{P_i\le t_r,S_i=r\}\mathrm{/}(r+1)$. Each event is measurable, and [F8] and [F10] give the absolute expectation of its summand as $P_{\theta }(P_i\le t_r,S_i=r)\mathrm{/}(r+1)\le 1$, so each summand is integrable. Then [F8], [F9], and step 2.1 give ${\mathbb{E}}_{\theta }[I_i\mathrm{/}(R\vee 1)]={\sum }_{r=0}^{m-1}{t}_r{P}_{\theta }(S_i=r)\mathrm{/}(r+1)=(q\mathrm{/}m){\sum }_{r=0}^{m-1}{P}_{\theta }(S_i=r)=q\mathrm{/}m$.

The measurable BH rejection rule from step 1.1 lets us apply [F1] with $P=P_{\theta }$ and $I_0=I_0(\theta )$. Thus $V={\sum }_{i\in I_0(\theta )}{I}_i$ and ${\mathrm{FDR}}_{\theta }={\mathbb{E}}_{\theta }[V\mathrm{/}(R\vee 1)]$; repeated finite linearity [F9] and step 3.1 give ${\mathrm{FDR}}_{\theta }={\sum }_{i\in I_0(\theta )}q\mathrm{/}m=(m_0(\theta )\mathrm{/}m)q\le q$, since $m_0(\theta )\le m$. This is the FDR defined in [F1], not an FWER conclusion.