SlowGuess/NeurIPS_2023_OCR huggingface dataset:
(i) For any measure  $\mathbb{Q}$  on  $\mathcal{H}_U$  and any causal mechanism  $\mathbb{L}$  on  $(\Omega, \mathcal{H}_U, \mathbb{Q})$ , the causal kernel  $K_U^{\mathrm{do}(U,\mathbb{Q},\mathbb{L})} = K_U$  is a version of  $\mathbb{P}_{\mathcal{H}_U}^{\mathrm{do}(U,\mathbb{Q})}$ , which means that  $\mathcal{H}_U$  is a global source  $\sigma$ -algebra of the interveneded causal space  $(\Omega, \mathcal{H}, \mathbb{P}^{\mathrm{do}(U,\mathbb{Q})}, \mathbb{K}^{\mathrm{do}(U,\mathbb{Q},\mathbb{L})})$ .
(ii) Suppose  $V \in \mathcal{P}(T)$  with  $V \subseteq U$ . Suppose that the measure  $\mathbb{Q}$  on  $(\Omega, \mathcal{H}_U)$  factorises over  $\mathcal{H}_V$  and  $\mathcal{H}_{U \setminus V}$ , i.e. for any  $A \in \mathcal{H}_V$  and  $B \in \mathcal{H}_{U \setminus V}$ ,  $\mathbb{Q}(A \cap B) = \mathbb{Q}(A) \mathbb{Q}(B)$ . Then after a hard intervention on  $\mathcal{H}_U$  via  $\mathbb{Q}$ , the causal kernel  $K_V^{\mathrm{do}(U, \mathbb{Q}, \mathrm{hard})}$  is a version of  $\mathbb{P}_V^{\mathrm{do}(U, \mathbb{Q})}$ , which means that  $\mathcal{H}_V$  is a global source  $\sigma$ -algebra of the interveneded causal space  $(\Omega, \mathcal{H}, \mathbb{P}^{\mathrm{do}(U, \mathbb{Q})}, \mathbb{K}^{\mathrm{do}(U, \mathbb{Q}, \mathrm{hard})})$ .
Let  $A \in \mathcal{H}$  be an event, and  $U \in \mathcal{P}(T)$ . By the definition of the intervention measure (Definition 2.3), we always have
$$
\mathbb {P} ^ {\mathrm {d o} (U, \mathbb {Q})} (A) = \int \mathbb {Q} (d \omega) K _ {U} (\omega , A),
$$
hence  $\mathbb{P}^{\mathrm{do}(U,\mathbb{Q})}(A)$  can be written in terms of  $\mathbb{P}$  and  $\mathbb{Q}$  if  $K_U(\omega, A)$  can be written in terms of  $\mathbb{P}$ . This can be seen to occur in three trivial cases: first, if  $\mathcal{H}_U$  is a local source of  $A$  (see Definition D.1), in which case  $K_U(\omega, A) = \mathbb{P}_{\mathcal{H}_U}(\omega, A)$ ; secondly, if  $\mathcal{H}_U$  has no causal effect on  $A$  (see Definition B.1), in which case  $K_U(\omega, A) = \mathbb{P}(A)$ ; and finally, if  $A \in \mathcal{H}_U$ , in which case, by intervention determinism (Definition 2.2(ii), we have  $K_U(\omega, A) = 1_A(\omega)$ . In the latter case, we do not even have dependence on  $\mathbb{P}$ . Can we generalise these results?
Lemma D.3. Let  $(\Omega, \mathcal{H}, \mathbb{P}, \mathbb{K}) = (\times_{t \in T} E_t, \otimes_{t \in T} \mathcal{E}_t, \mathbb{P}, \mathbb{K})$  be a causal space. Let  $A \in \mathcal{H}$  be an event, and  $U \in \mathcal{P}(T)$ . If there exists a sub- $\sigma$ -algebra  $\mathcal{G}$  of  $\mathcal{H}$  (not necessarily of the form  $\mathcal{H}_V$  for some  $V \in \mathcal{P}(T)$ ) such that
(i) the conditional probability  $\mathbb{P}_{\mathcal{H}_U \vee \mathcal{G}}^{\mathrm{do}(U, \mathbb{Q})}(\cdot, A)$  can be written in terms of  $\mathbb{P}$  and  $\mathbb{Q}$ ;
(ii) the causal kernel  $K_U(\cdot, B)$  can be written in terms of  $\mathbb{P}$  for all  $B \in \mathcal{G}$ ;
then  $\mathbb{P}^{\mathrm{do}(U,\mathbb{Q})}(A)$  can be written in terms of  $\mathbb{P}$  and  $\mathbb{Q}$ .
Remark D.4. The three cases discussed in the paragraph above Lemma D.3 are special cases of the Lemma with  $\mathcal{G}$  being any sub- $\sigma$ -algebra of  $\mathcal{H}$  with  $\{\emptyset, \Omega\} \subseteq \mathcal{G} \subseteq \mathcal{H}_U$ . In this case, condition (ii) is trivially satisfied since we have  $K_U(\cdot, B) = 1_B(\cdot)$  by intervention determinism (Definition 2.2(ii)), and for condition (i), by Theorem D.2(i), we have  $\mathbb{P}_{\mathcal{H}_U}^{\mathrm{do}(U, \mathbb{Q})}(\cdot, A) = K_U(\cdot, A)$ , which means that the problem reduces to checking if  $K_U(\cdot, A)$  can be written in terms of  $\mathbb{P}$ .
Proof. By law of total expectations, for any  $V \in \mathcal{P}(T)$ , we have
$$
\begin{array}{l} \mathbb {P} ^ {\mathrm {d o} (U, \mathbb {Q})} (A) = \int \mathbb {P} _ {\mathcal {H} _ {U} \vee \mathsf {G}} ^ {\mathrm {d o} (U, \mathbb {Q})} (\omega , A) \mathbb {P} ^ {\mathrm {d o} (U, \mathbb {Q})} (d \omega) \\ = \int \mathbb {P} _ {\mathcal {H} _ {U} \vee \mathcal {G}} ^ {\mathrm {d o} (U, \mathbb {Q})} (\omega , A) \int \mathbb {Q} (d \omega^ {\prime}) K _ {U} (\omega^ {\prime}, d \omega). \\ \end{array}
$$
Here,  $\mathbb{P}_{\mathcal{H}_U \vee \mathcal{G}}^{\mathrm{do}(U, \mathbb{Q})}(\omega, A)$  can be written in terms of  $\mathbb{P}$  and  $\mathbb{Q}$  by condition (i). Moreover, note that it suffices to be able to write the restriction of  $K_U(\omega', \cdot)$  to  $\mathcal{H}_U \vee \mathcal{G}$  in terms of  $\mathbb{P}$ , since the integration is of a  $\mathcal{H}_U \vee \mathcal{G}$ -measurable function. Since the collection of intersections  $\{D \cap B, D \in \mathcal{H}_U, B \in \mathcal{G}\}$  is a  $\pi$ -system that generates  $\mathcal{H}_U \vee \mathcal{G}$  [14, p.5, 1.18], it suffices to check that  $K_U(\omega', D \cap B)$  can be written in terms of  $\mathbb{P}$  for all  $D \in \mathcal{H}_U$  and  $B \in \mathcal{G}$ . But by interventional determinism (Definition 2.2(ii)), we have  $K_U(\omega', D \cap B) = 1_D(\omega') K_U(\omega', B)$ . Since  $K_U(\omega', B)$  can be written in terms of  $\mathbb{P}$  by condition (ii), the restriction of  $K_U(\omega', \cdot)$  to  $\mathcal{H}_U \vee \mathcal{G}$  can be written in terms of  $\mathbb{P}$ , and hence  $\mathbb{P}_{\mathcal{H}_U}^{\mathrm{do}(U, \mathbb{Q})}(A)$  can be written in terms of  $\mathbb{P}$  and  $\mathbb{Q}$ .
Corollary D.5. Let  $(\Omega, \mathcal{H}, \mathbb{P}, \mathbb{K}) = (\times_{t \in T} E_t, \otimes_{t \in T} \mathcal{E}_t, \mathbb{P}, \mathbb{K})$  be a causal space. Let  $A \in \mathcal{H}$  be an event, and  $U \in \mathcal{P}(T)$ . If there exists a  $V \in \mathcal{P}(T)$  such that condition (i) of Lemma D.3 is satisfied with  $\mathcal{G} = \mathcal{H}_V$  and one of the following conditions is satisfied:
(a)  $\mathcal{H}_U$  is a local source of  $\mathcal{H}_V$ ; or
(b)  $\mathcal{H}_U$  has no causal effect on  $\mathcal{H}_V$ ; or
(c)  $V\subseteq U$
Proof. Condition (i) of Lemma D.3 is satisfied by hypothesis. If one of (a), (b) or (c) is satisfied, then trivially, condition (ii) of Lemma D.3 is also satisfied. The result now follows from Lemma D.3.  $\square$
The above is reminiscent of "valid adjustments" in the context of structural causal models [46, p.115, Proposition 6.41], and in fact contains the valid adjustments.
E Counterfactuals
There are various notions of counterfactuals in the literature. The one considered in the SCM literature is the interventional counterfactual, which captures the notion of "what would have happened if we intervened on the space, given some observations (that are possibly contradictory to the intervention we imagine we would have done)". Recently, backtracking counterfactuals have also been integrated into the SCM framework [58]. This captures the notion of "what would have happened if background conditions of the world had been different, given that the causal laws of the system stay the same?" Finally, we note that in the potential outcomes framework, the random variables representing "potential outcomes" that form the primitives of the framework can be directly counterfactual.
Vanilla probability measures have just one argument, i.e. the event. Conditional measures and causal kernels (in the sense of our Definition 2.2) have two arguments, the first being the outcome which we either observe or force the occurrence of, and the second being the event in whose measure we are interested. For both of the above concepts of counterfactuals, we need to go one step further and consider three arguments. The first is the outcome which we observe, just like in conditioning, and the last should be the event in whose measure we are interested. For interventional counterfactuals, the second argument should be an outcome which we imagine to have forced the occurrence of given that we observed the outcome of the first argument, and for backtracking counterfactuals, the second argument should be an outcome which we imagine to have observed instead of the outcome in the first argument which we actually observed.
From these principles, we tentatively propose to extend Definition 2.2 to account for interventional counterfactuals as follows.
Definition E.1. A causal space is defined as the quadruple  $(\Omega, \mathcal{H}, \mathbb{P}, \mathbb{K})$ , where  $(\Omega, \mathcal{H}, \mathbb{P}) = (\times_{t \in T} E_t, \otimes_{t \in T} \mathcal{E}_t, \mathbb{P})$  is a probability space and  $\mathbb{K} = \{K_{S,\mathcal{F}} : S \in \mathcal{P}(T), \mathcal{F} \text{ sub-}\sigma$ -algebra of  $\mathcal{H}\}$ , called the causal mechanism, is a collection of functions  $K_{S,\mathcal{F}} : \Omega \times \Omega \times \mathcal{H} \to [0,1]$ , called the causal kernel on  $\mathcal{H}_S$  after observing  $\mathcal{F}$ , such that
(i) for each fixed  $\eta \in \Omega$  and  $A\in \mathcal{H}$ ,  $K_{S,\mathcal{F}}(\cdot ,\eta ,A)$  is measurable with respect to  $\mathcal{H}_S$ ;
(ii) for each fixed  $\omega \in \Omega$  and  $A\in \mathcal{H}$ ,  $K_{S,\mathcal{F}}(\omega ,\cdot ,A)$  is measurable with respect to  $\mathcal{F}$ ;
(iii) for each fixed pair  $(\omega, \eta) \in \Omega \times \Omega$ ,  $K_{S,\mathcal{F}}(\omega, \eta, \cdot)$  is a measure on  $\mathcal{H}$ ;
(iv) for all  $A\in \mathcal{H}$  and  $\omega ,\eta \in \Omega$
$$
K _ {\emptyset , \mathcal {F}} (\omega , \eta , A) = \mathbb {P} _ {\mathcal {F}} (\eta , A);
$$
(v) for all  $A\in \mathcal{H}_S$  , all  $B\in \mathcal{H}$  and all  $\omega ,\eta \in \Omega$
$$
K _ {S, \mathcal {F}} (\omega , \eta , A \cap B) = 1 _ {A} (\omega) K _ {S} (\omega , \eta , B);
$$
in particular, for  $A \in \mathcal{H}_S$ ,  $K_{S,\mathcal{F}}(\omega, \eta, A) = 1_A(\omega) K_{S,\mathcal{F}}(\omega, \eta, \Omega) = 1_A(\omega)$ ;
(vi) for all  $A\in \mathcal{H}$ $\omega \in \Omega$  and sub-  $\sigma$  -algebras  $\mathcal{F}\subseteq \mathfrak{G}\subseteq \mathcal{H}$
$$
\mathbb {E} _ {\mathcal {F}} \left[ K _ {S, \mathcal {G}} (\omega , \cdot , A) \right] = K _ {S, \mathcal {F}} (\omega , \cdot , A).
$$
Note that letting  $\mathcal{F} = \{\varnothing, \Omega\}$  trivially recovers the causal space as defined in Definition 2.2. Moreover, letting  $S = \emptyset$ , we recover the conditional distribution given  $\mathcal{F}$ .
Recall that the one of the biggest philosophical differences between the SCM framework and our proposed causal spaces (Definition 2.2) was the fact that SCMs start with the variables, the structural equations and the noise distributions as the primitive objects, and the observational and interventional distributions over the endogenous variables are derived from these, whereas causal spaces take the observational and interventional distributions as the primitive objects (the latter via causal kernels). Note that, in the above extended definition of causal spaces incorporating interventional counterfactuals (Definition E.1), we applied the same principles, in that we treated the observational distribution  $(\mathbb{P})$ , interventional distributions  $(K_{S,\{\emptyset ,\Omega \}})$  and the (interventional) counterfactual distributions  $(K_{S,\mathcal{F}})$  as the primitive objects.
This differs significantly from the SCM framework, where again, the (interventional) counterfactual distributions are derived from the structural equations, by first conditioning on the observed values of
the endogenous variables to get a modified (often Dirac) measure on the exogenous variables, then intervening on some of the endogenous variables, deriving the measure on the rest of the endogenous variables by propagating these through the same structural equations. We see the value in this approach in that the (interventional) counterfactual distributions can be neatly derived from the same primitive objects that are used to calculate the observational and interventional distribution. However, we argue that this cannot be an axiomatisation of (interventional) counterfactual distributions in the strictest sense, because it relies on assumptions. In particular, it strongly relies on the assumption that the endogenous variables have no causal effect on the exogenous variables, and when this assumption is violated, i.e. when there is a hidden mediator, calculation of (interventional) counterfactual distributions is not possible. In contrast, Definition E.1 treat the (interventional) counterfactual measures as the primitive objects, and does not impose any a priori assumptions about the system.
As mentioned in Section 5 of the main body of the paper, we leave further developments of this interventional counterfactual causal space, as well as the definition of backtracking counterfactual causal space, as essential future work.
F Proofs
Proof. That  $\mathbb{P}^{\mathrm{do}(U,\mathbb{Q})}$  is a measure on  $(\Omega ,\mathcal{H})$  follows immediately from the usual construction of measures from measures and transition probability kernels, see e.g. Cinar [14, p.38, Theorem 6.3]. It remains to check that  $\mathbb{K}^{\mathrm{do}(U,\mathbb{Q},\mathbb{L})}$  is a valid causal mechanism in the sense of Definition 2.2.
(i) For all  $A\in \mathcal{H}$  and  $\omega \in \Omega$
$$
\begin{array}{l} K _ {\emptyset} ^ {\mathrm {d o} (U, \mathbb {Q}, \mathbb {L})} (\omega , A) = \int L _ {\emptyset} \left(\omega_ {\emptyset}, d \omega_ {U} ^ {\prime}\right) K _ {U} \left(\left(\omega_ {\emptyset}, \omega_ {U} ^ {\prime}\right), A\right) \\ = \int \mathbb {Q} (d \omega^ {\prime}) K _ {U} (\omega^ {\prime}, A) \\ = \mathbb {P} ^ {\mathrm {d o} (U, \mathbb {Q})} (A), \\ \end{array}
$$
where we applied Axiom 2.2(i) to  $L_{\varnothing}$ .
(ii) For all  $A \in \mathcal{H}_S$  and  $B \in \mathcal{H}$ , we have, by Axiom 2.2(ii) using the fact that  $A \in \mathcal{H}_S \subseteq \mathcal{H}_{S \cup U}$ ,
$$
\begin{array}{l} K _ {S} ^ {\mathrm {d o} (U, \mathbb {Q}, \mathbb {L})} (\omega , A \cap B) \\ = \int L _ {S \cap U} (\omega_ {S \cap U}, d \omega_ {U} ^ {\prime}) K _ {S \cup U} ((\omega_ {S \setminus U}, \omega_ {U} ^ {\prime}), A \cap B) \\ = \int L _ {S \cap U} \left(\omega_ {S \cap U}, d \omega_ {U} ^ {\prime}\right) 1 _ {A} \left(\left(\omega_ {S \setminus U}, \omega_ {U} ^ {\prime}\right)\right) K _ {S \cup U} \left(\left(\omega_ {S \setminus U}, \omega_ {U} ^ {\prime}\right), B\right) \\ = \int L _ {S \cap U} (\omega_ {S \cap U}, d \omega_ {U} ^ {\prime}) 1 _ {A} ((\omega_ {S \setminus U}, \omega_ {S \cap U} ^ {\prime})) K _ {S \cup U} ((\omega_ {S \setminus U}, \omega_ {U} ^ {\prime}), B), \\ \end{array}
$$
where, in going from the third line to the fourth, we split the  $\omega_U'$  in  $1_A((\omega_{S\setminus U},\omega_U'))$  into components  $(\omega_{S\cap U}^{\prime},\omega_{U\setminus S}^{\prime})$  and notice that since  $A\in \mathcal{H}_S$ ,  $1_A$  does not depend on the component  $\omega_{U\setminus S}^{\prime}$ . Here, the map  $\omega_{S\cap U}^{\prime}\mapsto 1_A((\omega_{S\setminus U},\omega_{S\cap U}^{\prime}))$  is  $\mathcal{H}_{S\cap U}$ -measurable, so we can write it as the limit of an increasing sequence of positive  $\mathcal{H}_{S\cap U}$ -simple functions (see Section A.1), say  $(f_n)_{n\in \mathbb{N}}$  with  $f_{n} = \sum_{i_{n} = 1}^{m_{n}}b_{i_{n}}1_{B_{i_{n}}}$ , where  $B_{i_n}\in \mathcal{H}_{S\cap U}$ . Likewise, the map  $\omega_U^\prime \mapsto K_{S\cup U}((\omega_{S\setminus U},\omega_U'),B)$  is  $\mathcal{H}_U$ -measurable, so we can write it as the limit of an increasing sequence of positive  $\mathcal{H}_U$ -simple functions, say  $(g_n)_{n\in \mathbb{N}}$  with  $g_{n} = \sum_{j_{n} = 1}^{l_{n}}c_{j_{n}}1_{C_{j_{n}}}$ , where  $C_{j_n}\in \mathcal{H}_U$ . Hence
$$
K _ {S} ^ {\mathrm {d o} (U, \mathbb {Q}, \mathbb {L})} (\omega , A \cap B) = \int L _ {S \cap U} (\omega_ {S \cap U}, d \omega_ {U} ^ {\prime}) \left(\lim _ {n \to \infty} f _ {n} (\omega_ {S \cap U} ^ {\prime})\right) \left(\lim _ {n \to \infty} g _ {n} (\omega_ {U} ^ {\prime})\right).
$$
Since, for each  $\omega_U'$ , both of the limits exist by construction, namely the original measurable functions, we have that the product of the limits is the limit of the products:
$$
K _ {S} ^ {\mathrm {d o} (U, \mathbb {Q}, \mathbb {L})} (\omega , A \cap B) = \int L _ {S \cap U} (\omega_ {S \cap U}, d \omega_ {U} ^ {\prime}) \lim _ {n \to \infty} \big (f _ {n} (\omega_ {S \cap U} ^ {\prime}) g _ {n} (\omega_ {U} ^ {\prime}) \big).
$$
Here, since  $f_{n}$  and  $g_{n}$  were individually sequences of increasing functions, the pointwise products  $f_{n}g_{n}$  also form an increasing sequence of functions. Hence, we can apply the monotone convergence theorem to see that
$$
\begin{array}{l} K _ {S} ^ {\mathrm {d o} (U, \mathbb {Q}, \mathbb {L})} (\omega , A \cap B) \\ = \lim  _ {n \rightarrow \infty} \int L _ {S \cap U} \left(\omega_ {S \cap U}, d \omega_ {U} ^ {\prime}\right) f _ {n} \left(\omega_ {S \cap U} ^ {\prime}\right) g _ {n} \left(\omega_ {U} ^ {\prime}\right) \\ = \lim  _ {n \rightarrow \infty} \sum_ {i _ {n} = 1} ^ {m _ {n}} \sum_ {j _ {n} = 1} ^ {l _ {n}} b _ {i _ {n}} c _ {j _ {n}} \int L _ {S \cap U} \left(\omega_ {S \cap U}, d \omega_ {U} ^ {\prime}\right) 1 _ {B _ {i _ {n}}} \left(\omega_ {S \cap U} ^ {\prime}\right) 1 _ {C _ {j _ {n}}} \left(\omega_ {U} ^ {\prime}\right) \\ = \lim  _ {n \rightarrow \infty} \sum_ {i _ {n} = 1} ^ {m _ {n}} \sum_ {j _ {n} = 1} ^ {l _ {n}} b _ {i _ {n}} c _ {j _ {n}} L _ {S \cap U} \left(\omega_ {S \cap U}, B _ {i _ {n}} \cap C _ {j _ {n}}\right) \\ = \lim  _ {n \rightarrow \infty} \sum_ {i _ {n} = 1} ^ {m _ {n}} \sum_ {j _ {n} = 1} ^ {l _ {n}} b _ {i _ {n}} c _ {j _ {n}} 1 _ {B _ {i _ {n}}} (\omega_ {S \cap U}) L _ {S \cap U} (\omega_ {S \cap U}, C _ {j _ {n}}) \\ = \lim  _ {n \rightarrow \infty} \sum_ {i _ {n} = 1} ^ {m _ {n}} b _ {i _ {n}} 1 _ {B _ {i _ {n}}} \left(\omega_ {S \cap U}\right) \sum_ {j _ {n} = 1} ^ {l _ {n}} c _ {j _ {n}} L _ {S \cap U} \left(\omega_ {S \cap U}, C _ {j _ {n}}\right) \\ = \left(\lim  _ {n \rightarrow \infty} \sum_ {i _ {n} = 1} ^ {m _ {n}} b _ {i _ {n}} 1 _ {B _ {i _ {n}}} \left(\omega_ {S \cap U}\right)\right)\left(\lim  _ {n \rightarrow \infty} \sum_ {j _ {n} = 1} ^ {l _ {n}} c _ {j _ {n}} L _ {S \cap U} \left(\omega_ {S \cap U}, C _ {j _ {n}}\right)\right) \\ = \left(\lim  _ {n \rightarrow \infty} f _ {n} \left(\omega_ {S \cap U}\right)\right)\left(\lim  _ {n \rightarrow \infty} \int L _ {S \cap U} \left(\omega_ {S \cap U}, d \omega_ {U} ^ {\prime}\right) \sum_ {j _ {n} = 1} ^ {l _ {n}} c _ {j} 1 _ {C _ {j _ {n}}} \left(\omega_ {U} ^ {\prime}\right)\right) \\ = 1 _ {A} \left(\left(\omega_ {S \backslash U}, \omega_ {S \cap U}\right)\right) \int L _ {S \cap U} \left(\omega_ {S \cap U}, d \omega_ {U} ^ {\prime}\right) \lim  _ {n \rightarrow \infty} g _ {n} \left(\omega_ {U} ^ {\prime}\right) \\ = 1 _ {A} \left(\omega_ {S}\right) \int L _ {S \cap U} \left(\omega_ {S \cap U}, d \omega_ {U} ^ {\prime}\right) K _ {S \cup U} \left(\left(\omega_ {S \setminus U}, \omega_ {U} ^ {\prime}\right), B\right) \\ = 1 _ {A} \left(\omega_ {S}\right) K _ {S} ^ {\mathrm {d o} (U, \mathbb {Q}, \mathbb {L})} \left(\omega_ {S}, B\right) \\ \end{array}
$$
where, from the fourth line to the fifth, we used Axiom 2.2(ii); from the sixth line to the seventh, we used that limit of the products is the product of the limits again, noting that both of the limits exist by construction; from the eighth line to the ninth, we used monotone convergence theorem again. This is the required result.
Theorem C.3. Let  $(\Omega, \mathcal{H}, \mathbb{P}, \mathbb{K}) = (\times_{t \in T} E_t, \otimes_{t \in T} \mathcal{E}_t, \mathbb{P}, \mathbb{K})$  be a causal space, and  $U \in \mathcal{P}(T)$  and  $\mathbb{Q}$  a probability measure on  $(\Omega, \mathcal{H}_U)$ . Then after a hard intervention on  $\mathcal{H}_U$  via  $\mathbb{Q}$ , the intervention causal kernels  $K_S^{\mathrm{do}(U,\mathbb{Q},\mathrm{hard})}$  are given by
Proof. We decompose  $\mathcal{H}_U$  as a product  $\sigma$ -algebra into  $\mathcal{H}_{S\cap U}\otimes \mathcal{H}_{U\setminus S}$ . Then events of the form  $B\cap C$  with  $B\in \mathcal{H}_{S\cap U}$  and  $C\in \mathcal{H}_{U\setminus S}$  generate  $\mathcal{H}_U$ , so for fixed  $\omega_{S\cap U}$ , the measure  $L_{S\cap U}(\omega_{S\cap U},\cdot)$  is completely determined by  $L_{S\cap U}(\omega_{S\cap U},B\cap C)$  for all  $B\in \mathcal{H}_{S\cap U}$ ,  $C\in \mathcal{H}_{U\setminus S}$ . But we have
$$
L _ {S \cap U} \left(\omega_ {S \cap U}, B \cap C\right) = \delta_ {\omega_ {S \cap U}} (B) L _ {S \cap U} \left(\omega_ {S \cap U}, C\right) \quad \text {b y}
$$
$$
= \delta_ {\omega_ {S \cap U}} (B) \mathbb {Q} (C),
$$
since  $L_{S\cap U}$  is trivial and  $C\in \mathcal{H}_{U\setminus S}$ . So the measure  $L_{S\cap U}(\omega_{S\cap U},\cdot)$  is a product measure of  $\delta_{\omega_{S\cap U}}$  and  $\mathbb{Q}$ . Hence, applying Fubini's theorem,
$$
\begin{array}{l} K _ {S} ^ {\mathrm {d o} (U, \mathbb {Q}, \mathrm {h a r d})} (\omega , A) = \int L _ {S \cap U} (\omega_ {S \cap U}, d \omega_ {U} ^ {\prime}) K _ {S \cup U} ((\omega_ {S \setminus U}, \omega_ {U} ^ {\prime}), A) \\ = \int \int K _ {S \cup U} ((\omega_ {S \setminus U}, \omega_ {S \cap U} ^ {\prime}, \omega_ {U \setminus S} ^ {\prime}), A) \delta_ {\omega_ {S \cap U}} (d \omega_ {S \cap U} ^ {\prime}) \mathbb {Q} (d \omega_ {U \setminus S} ^ {\prime}) \\ = \int K _ {S \cup U} \left(\left(\omega_ {S \backslash U}, \omega_ {S \cap U}, \omega_ {U \backslash S} ^ {\prime}\right), A\right) \mathbb {Q} \left(d \omega_ {U \backslash S} ^ {\prime}\right) \\ = \int \mathbb {Q} (d \omega_ {U \setminus S} ^ {\prime}) K _ {S \cup U} ((\omega_ {S}, \omega_ {U \setminus S} ^ {\prime}), A), \\ \end{array}
$$
(i) For any measure  $\mathbb{Q}$  on  $\mathcal{H}_U$  and any causal mechanism  $\mathbb{L}$  on  $(\Omega, \mathcal{H}_U, \mathbb{Q})$ , the causal kernel  $K_U^{\mathrm{do}(U,\mathbb{Q},\mathbb{L})} = K_U$  is a version of  $\mathbb{P}_{\mathcal{H}_U}^{\mathrm{do}(U,\mathbb{Q})}$ , which means that  $\mathcal{H}_U$  is a global source  $\sigma$ -algebra of the interveneded causal space  $(\Omega, \mathcal{H}, \mathbb{P}^{\mathrm{do}(U,\mathbb{Q})}, \mathbb{K}^{\mathrm{do}(U,\mathbb{Q},\mathbb{L})})$ .
(ii) Suppose  $V \in \mathcal{P}(T)$  with  $V \subseteq U$ . Suppose that the measure  $\mathbb{Q}$  on  $(\Omega, \mathcal{H}_U)$  factorises over  $\mathcal{H}_V$  and  $\mathcal{H}_{U \setminus V}$ , i.e. for any  $A \in \mathcal{H}_V$  and  $B \in \mathcal{H}_{U \setminus V}$ ,  $\mathbb{Q}(A \cap B) = \mathbb{Q}(A) \mathbb{Q}(B)$ . Then after a hard intervention on  $\mathcal{H}_U$  via  $\mathbb{Q}$ , the causal kernel  $K_V^{\mathrm{do}(U, \mathbb{Q})}$  is a version of  $\mathbb{P}_V^{\mathrm{do}(U, \mathbb{Q})}$ , which means that  $\mathcal{H}_V$  is a global source  $\sigma$ -algebra of the intervened causal space  $(\Omega, \mathcal{H}, \mathbb{P}^{\mathrm{do}(U, \mathbb{Q})}, \mathbb{K}^{\mathrm{do}(U, \mathbb{Q})})$ .
Proof. Suppose that  $f = \sum_{i=1}^{m} b_i 1_{B_i}$  is a  $\mathcal{H}_U$ -simple function, i.e. with  $B_i \in \mathcal{H}_U$  for  $i = 1, \dots, m$ . Then for any  $B \in \mathcal{H}_U$ ,
$$
\begin{array}{l} \int_ {B} f (\omega) \mathbb {P} ^ {\mathrm {d o} (U, \mathbb {Q})} (d \omega) = \int_ {B} \sum_ {i = 1} ^ {m} b _ {i} 1 _ {B _ {i}} (\omega) \mathbb {P} ^ {\mathrm {d o} (U, \mathbb {Q})} (d \omega) \\ = \sum_ {i = 1} ^ {m} b _ {i} \mathbb {P} ^ {\mathrm {d o} (U, \mathbb {Q})} (B \cap B _ {i}) \\ = \sum_ {i = 1} ^ {m} b _ {i} \int \mathbb {Q} (d \omega) K _ {U} (\omega , B \cap B _ {i}) \quad \text {b y} \\ = \sum_ {i = 1} ^ {m} b _ {i} \int \mathbb {Q} (d \omega) 1 _ {B \cap B _ {i}} (\omega) \quad \text {b y} \\ = \int_ {B} \sum_ {i = 1} ^ {m} b _ {i} 1 _ {B _ {i}} (\omega) \mathbb {Q} (d \omega) \\ = \int_ {B} f (\omega) \mathbb {Q} (d \omega). \\ \end{array}
$$
Now, for any  $\mathcal{H}_U$ -measurable map  $g: \Omega \to \mathbb{R}$ , we can write it as a limit of an increasing sequence of positive  $\mathcal{H}_U$ -simple functions  $f_n$  (see Section A.1), so for any  $B \in \mathcal{H}_U$ ,