Proof of Theorem 3.1 from a measure theoretical perspective
*(In this comment, we give some details on how to prove Theorem 3.1 of our submission from a purely measure theoretical perspective, using the condition above)*
The joint distribution over graphs & parameters is defined over the measurable union space $\mathcal{X} \triangleq \bigcup_{G\in\mathbf{G}}(\{G\}\times \Theta_{G})$, where $\mathbf{G}$ represents the (finite) set of all DAGs over $d$ nodes. We first can prove that if we have transitions $G \rightarrow G' \rightarrow G''$ in the GFlowNet (i.e., $G'$ is the result of adding one edge to $G$, and $G''$ is the result of adding one edge to $G'$), then for any bounded measurable function $h: \Theta_{G}\times \Theta_{G''}\rightarrow \mathbb{R}$, we have:
$$
\iint_{\Theta_{G}\times \Theta_{G''}}h(\theta, \theta'')R(G'', d\theta'')P_{B}(G\mid G')P_{B}(G'\mid G'')P_{\phi}(d\theta\mid G) = \iint_{\Theta_{G}\times \Theta_{G''}}h(\theta, \theta'')R(G, d\theta)P_{\phi}(G''\mid G')P_{\phi}(G'\mid G)P_{\phi}(d\theta''\mid G'')
$$
We can use the SubTB condition above, observing that $h(\theta, \theta'')$ is a bounded measurable function of $\Theta_{G'}\times \Theta_{G''}$ (constant on $\Theta_{G'}$ for a fixed $\theta$), and using Fubini-Tonelli’s theorem.
$$\Bigg[\iint_{\Theta_{G}\times \Theta_{G''}} h(\theta, \theta'')R(G'', d\theta'')P_{B}(G\mid G')P_{B}(G'\mid G'')P_{\phi}(d\theta\mid G)\Bigg]\times\Bigg[\int_{\Theta_{G'}}P_{\phi}(d\theta'\mid G')\Bigg]$$
$$= \int_{\Theta_{G}}\Bigg[\iint_{\Theta_{G'}\times\Theta_{G''}}h(\theta, \theta'')R(G'', d\theta'')P_{B}(G'\mid G'')P_{\phi}(d\theta'\mid G')\Bigg]P_{B}(G\mid G')P_{\phi}(d\theta\mid G)$$
$$= \int_{\Theta_{G}}\Bigg[\iint_{\Theta_{G'}\times\Theta_{G''}}h(\theta, \theta'')R(G', d\theta')P_{\phi}(G''\mid G')P_{\phi}(d\theta''\mid G'')\Bigg]P_{B}(G\mid G')P_{\phi}(d\theta\mid G)$$
$$= \iint_{\Theta_{G}\times \Theta_{G'}}\Bigg[\int_{\Theta_{G''}}h(\theta, \theta'')P_{\phi}(G''\mid G')P_{\phi}(d\theta''\mid G'')\Bigg]R(G', d\theta')P_{B}(G\mid G')P_{\phi}(d\theta\mid G)$$
The quantity inside the brackets is a bounded measurable function of $\Theta_{G}\times \Theta_{G'}$. We can therefore apply the SubTB condition above.
$$= \iint_{\Theta_{G}\times\Theta_{G'}}\Bigg[\int_{\Theta_{G''}}h(\theta, \theta'')P_{\phi}(G''\mid G')P_{\phi}(d\theta''\mid G'')\Bigg]R(G, d\theta)P_{\phi}(G'\mid G)P_{\phi}(d\theta'\mid G')$$
$$= \Bigg[\iint_{\Theta_{G}\times\Theta_{G''}}h(\theta, \theta'')R(G, d\theta)P_{\phi}(G'\mid G)P_{\phi}(G''\mid G')P_{\phi}(d\theta''\mid G'')\Bigg]\times\Bigg[\int_{\Theta_{G'}}P_{\phi}(d\theta'\mid G')\Bigg]$$
Which proves the equality above on the product space $\Theta_{G}\times\Theta_{G''}$. By induction, we can also prove that for a partial trajectory $G_{0} \rightarrow G_{1}\rightarrow \ldots \rightarrow G_{T}$ in the GFlowNet, we have for any bounded measurable function $h: \Theta_{G_{0}}\times \Theta_{G_{T}}\rightarrow \mathbb{R}$:
$$
\iint_{\Theta_{G_{0}}\times\Theta_{G_{T}}}h(\theta_{0}, \theta_{T})R(G_{T}, d\theta_{T})\prod_{t=0}^{T-1}P_{B}(G_{t}\mid G_{t+1})P_{\phi}(d\theta_{0}\mid G_{0}) = \iint_{\Theta_{G_{0}}\times\Theta_{G_{T}}}h(\theta_{0}, \theta_{T})R(G_{0}, d\theta_{0})\prod_{t=0}^{T-1}P_{\phi}(G_{t+1}\mid G_{t})P_{\phi}(d\theta_{T}\mid G_{T})
$$
Theorem 3.1 of our submission can be rewritten in measure theoretical terms as
> If the SubTB conditions above are satisfied for all undirected paths of length 3 between
any $(G, \theta)$ and $G', \theta')$ of the form $(G, \theta) \leftarrow (G, \cdot) \rightarrow (G', \cdot) \rightarrow (G', \theta')$, then we have for any bounded measurable function $h: \Theta_{G}\rightarrow \mathbb{R}$
> $\displaystyle \int_{\Theta_{G}}h(\theta)P_{\phi}^{\top}(G, d\theta) \triangleq \int_{\Theta_{G}}h(\theta) \sum_{G_{0}\rightsquigarrow G}\prod_{t=0}^{T-1}P_{\phi}(G_{t+1}\mid G_{t})P_{\phi}(d\theta\mid G)\propto \int_{\Theta_{G}}h(\theta)R(G, d\theta)$
And we can prove this using the lemma above. Since any bounded measurable function $h:\Theta_{G}\rightarrow \mathbb{R}$ is also a measurable function over $\Theta_{G_{0}}\times \Theta_{G}$ (it is constant wrt. $\theta_{0}$), we can directly apply the lemma above (using the notation $G = G_{T}$)
$$\Bigg[\int_{\Theta_{G_{0}}}R(G_{0}, d\theta_{0})\Bigg]\times \Bigg[\int_{\Theta_{G}}h(\theta)P_{\phi}^{\top}(G,d\theta)\Bigg]$$
$$= \sum_{G_{0}\rightsquigarrow G}\iint_{\Theta_{G_{0}}\times\Theta_{G}}h(\theta)R(G_{0}, d\theta_{0})\prod_{t=0}^{T-1}P_{\phi}(G_{t+1}\mid G_{t})P_{\phi}(d\theta\mid G)$$
$$= \sum_{G_{0}\rightsquigarrow G}\iint_{\Theta_{G_{0}}\times\Theta_{G}}h(\theta)R(G, d\theta)\prod_{t=0}^{T-1}P_{B}(G_{t}\mid G_{t+1})P_{\phi}(d\theta_{0}\mid G_{0})$$
$$= \Bigg[\int_{\Theta_{G_{0}}}P_{\phi}(d\theta_{0}\mid G_{0})\Bigg]\times \Bigg[\int_{\Theta_{G}}h(\theta)R(G, d\theta)\sum_{G_{0}\rightsquigarrow G}\prod_{t=0}^{T-1}P_{B}(G_{t}\mid G_{t+1})\Bigg]$$
$$= \int_{\Theta_{G}}h(\theta)R(G, d\theta)$$