SlowGuess/NeurIPS_2023_OCR huggingface dataset:
We say that  $\mathcal{H}_U$  has an active causal effect on  $\mathcal{F}$ , or that  $\mathcal{H}_U$  is actively causal to  $\mathcal{F}$ , if  $\mathcal{H}_U$  has an active causal effect on some  $A \in \mathcal{F}$ .
(iii) Otherwise, we say that  $\mathcal{H}_U$  has a dormant causal effect on  $A$ , or that  $\mathcal{H}_U$  is dormantly causal to  $A$ .
We say that  $\mathcal{H}_U$  has a dormant causal effect on  $\mathcal{F}$ , or that  $\mathcal{H}_U$  is dormantly causal to  $\mathcal{F}$ , if  $\mathcal{H}_U$  does not have an active causal effect on any event in  $\mathcal{F}$  and there exists  $A \in \mathcal{F}$  on which  $\mathcal{H}_U$  has a dormant causal effect.
Sometimes, we will say that  $\mathcal{H}_U$  has a causal effect on  $A$  to mean that  $\mathcal{H}_U$  has either an active or a dormant causal effect on  $A$ .
The intuition is as follows. For any  $S \in \mathcal{P}(T)$  and any fixed event  $A \in \mathcal{H}$ , consider the function  $\omega_{S} \mapsto K_{S}((\omega_{S \cap U}, \omega_{S \setminus U}), A)$ . If  $\mathcal{H}_{U}$  has no causal effect on  $A$ , then it means that the causal kernel does not depend on the  $\omega_{S \cap U}$  component of  $\omega_{S}$ . Since this has to hold for all  $S \in \mathcal{P}(T)$ , it means that it is possible to have, for example,  $K_{U}(\omega, A) = \mathbb{P}(A)$  for all  $\omega \in \Omega$  and yet for  $\mathcal{H}_{U}$  to have a causal effect on  $A$ . This would be precisely the case where  $\mathcal{H}_{U}$  has a dormant causal effect on  $A$ , and it means that, for some  $S \in \mathcal{P}(T)$ ,  $\omega_{S} \mapsto K_{S}((\omega_{S \cap U}, \omega_{S \setminus U}, A)$  does depend on the  $\omega_{S \cap U}$  component.
We collect some straightforward but important special cases in the following remark.
Remark B.2. (a) If  $\mathcal{H}_U$  has no causal effect on  $A$ , then letting  $S = U$  in Definition B.1(i) and applying Definition 2.2(i), we can see that, for all  $\omega \in \Omega$ ,
$$
K _ {U} (\omega , A) = K _ {U \backslash U} (\omega , A) = K _ {\emptyset} (\omega , A) = \mathbb {P} (A).
$$
In particular, this means that  $\mathcal{H}_U$  cannot have both no causal effect and active causal effect on  $A$ .
(b) It is immediate that the trivial  $\sigma$ -algebra  $\mathcal{H}_{\emptyset} = \{\emptyset, \Omega\}$  has no causal effect on any event  $A \in \mathcal{H}$ . Conversely, it is also clear that  $\mathcal{H}_U$  for any  $U \in \mathcal{P}(T)$  has no causal effect on the trivial  $\sigma$ -algebra.
(c) Let  $U \in \mathcal{P}(T)$  and  $\mathcal{F}$  a sub- $\sigma$ -algebra of  $\mathcal{H}$ . If  $\mathcal{H}_U \cap \mathcal{F} \neq \{\emptyset, \Omega\}$ , then  $\mathcal{H}_U$  has an active causal effect on  $\mathcal{F}$ , since, for  $A \in \mathcal{H}_U \cap \mathcal{F}$  with  $A \neq \emptyset$  and  $A \neq \Omega$ , Definition 2.2(ii) tells us that  $K_U(\cdot, A) = 1_A(\cdot) \neq \mathbb{P}(A)$ . In particular,  $\mathcal{H}_U$  has an active causal effect on itself. Further, the full  $\sigma$ -algebra  $\mathcal{H} = \mathcal{H}_T$  has an active causal effect on all of its sub- $\sigma$ -algebras except the trivial  $\sigma$ -algebra, and every  $\mathcal{H}_U, U \in \mathcal{P}(T)$  except the trivial  $\sigma$ -algebra has an active causal effect on the full  $\sigma$ -algebra  $\mathcal{H}$ .
(d) Let  $U \in \mathcal{P}(T)$  and  $\mathcal{F}_1, \mathcal{F}_2$  be sub- $\sigma$ -algebras of  $\mathcal{H}$ . If  $\mathcal{F}_1 \subseteq \mathcal{F}_2$  and  $\mathcal{H}_U$  has no causal effect on  $\mathcal{F}_2$ , then it is clear that  $\mathcal{H}_U$  has no causal effect on  $\mathcal{F}_1$ .
(e) If  $\mathcal{H}_U$  has no causal effect on an event  $A$ , then for any  $V \in \mathcal{P}(T)$  with  $V \subseteq U$ ,  $\mathcal{H}_V$  has no causal effect on  $A$ . Indeed, take any  $S \in \mathcal{P}(T)$ . Then using the fact that  $\mathcal{H}_U$  has no causal effect on  $A$ , see that, for any  $\omega \in \Omega$ ,
$$
K _ {S \backslash V} (\omega , A) = K _ {(S \backslash V) \backslash U} (\omega , A) \quad \text {a p p l y i n g D e f i n i t i o n B . 1 (i) w i t h S \backslash V}
$$
$$
\begin{array}{l} = K _ {S \backslash U} (\omega , A) \quad \text {s i n c e} V \subseteq U \\ = K _ {S} (\omega , A) \quad \text {a p p l y i n g D e f i n i t i o n B . 1 (i) w i t h S}. \\ \end{array}
$$
Since  $S \in \mathcal{P}(T)$  was arbitrary, we have that  $\mathcal{H}_V$  has no causal effect on  $A$ .
(f) Contrapositively, if  $U, V \in \mathcal{P}(T)$  with  $V \subseteq U$  and  $\mathcal{H}_V$  has a causal effect on  $A$ , then  $\mathcal{H}_U$  has a causal effect on  $A$ .
(g) If  $U\in \mathcal{P}(T)$  has no causal effect on  $A$  , then for any  $V\in \mathcal{P}(T)$  , we have
$$
K _ {V} (\omega , A) = K _ {U \cup V} (\omega , A).
$$
Indeed,
$$
\begin{array}{l} K _ {U \cup V} (\omega , A) = K _ {(U \cup V) \backslash (U \backslash V)} (\omega , A) \quad \text {s i n c e} U \backslash V \text {h a s n o c a u s a l e f f e c t o n} A \text {b y (e)} \\ = K _ {V} (\omega , A) \quad \text {s i n c e} (U \cup V) \backslash (U \backslash V) = V. \\ \end{array}
$$
(h) If  $U, V \in \mathcal{P}(T)$  and neither  $\mathcal{H}_U$  nor  $\mathcal{H}_V$  has a causal effect on  $A$ , then  $\mathcal{H}_{U \cup V}$  has no causal effect on  $A$ . Indeed, for any  $S \in \mathcal{P}(T)$  and any  $\omega \in \Omega$ ,
$$
\begin{array}{l} K _ {S \backslash (U \cup V)} (\omega , A) = K _ {(S \backslash U) \backslash V} (\omega , A) \\ = K _ {S \backslash U} (\omega , A) \quad \text {a s} V \text {h a s n o c a u s a l e f f e c t o n} A \\ = K _ {S} (\omega , A) \quad \text {a s} U \text {h a s n o c a u s a l e f f e c t o n} A. \\ \end{array}
$$
Since  $S \in \mathcal{P}(T)$  was arbitrary,  $\mathcal{H}_{U \cup V}$  has no causal effect on  $A$ .
(i) Contrapositively, if  $U,V\in \mathcal{P}(T)$  and  $\mathcal{H}_{U\cup V}$  has a causal effect on  $A$  , then either  $\mathcal{H}_U$  or  $\mathcal{H}_V$  has a causal effect on  $A$
Following the definition of no causal effect, we define the notion of a trivial causal kernel.
Definition B.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  $U \in \mathcal{P}(T)$ . We say that the causal kernel  $K_U$  is trivial if  $\mathcal{H}_U$  has no causal effect on  $\mathcal{H}_{T \setminus U}$ .
Note that we can decompose  $\mathcal{H}$  as  $\mathcal{H} = \mathcal{H}_U\otimes \mathcal{H}_{T\setminus U}$ , and so  $\mathcal{H}$  is generated by events of the form  $A\times B$  for  $A\in \mathcal{H}_U$  and  $B\in \mathcal{H}_{T\setminus U}$ . But if  $K_{U}$  is trivial, then we have, by Axiom 2.2(ii),  $K_U(\omega ,A\times B) = 1_A(\omega)\mathbb{P}(B)$  for such a rectangle.
We also define a "conditional" version of causal effects.
Definition B.4. 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,  $U, V \in \mathcal{P}(T)$ ,  $A \in \mathcal{H}$  an event and  $\mathcal{F}$  a sub- $\sigma$ -algebra of  $\mathcal{H}$  (not necessarily of the form  $\mathcal{H}_S$  for some  $S \in \mathcal{P}(T)$ ).
(i) If  $K_{S \cup V}(\omega, A) = K_{(S \cup V) \setminus (U \setminus V)}(\omega, A)$  for all  $S \in \mathcal{P}(T)$  and all  $\omega \in \Omega$ , then we say that  $\mathcal{H}_U$  has no causal effect on  $A$  given  $\mathcal{H}_V$ , or that  $\mathcal{H}_U$  is non-causal to  $A$  given  $\mathcal{H}_V$ .
We say that  $\mathcal{H}_U$  has no causal effect on  $\mathcal{F}$  given  $\mathcal{H}_V$ , or that  $\mathcal{H}_U$  is non-causal to  $\mathcal{F}$  given  $\mathcal{H}_V$ , if, for all  $A \in \mathcal{F}$ ,  $\mathcal{H}_U$  has no causal effect on  $A$  given  $\mathcal{H}_V$ .
(ii) If there exists  $\omega \in \Omega$  such that  $K_{U\cup V}(\omega ,A)\neq K_V(\omega ,A)$ , then we say that  $\mathcal{H}_U$  has an active causal effect on  $A$  given  $\mathcal{H}_V$ , or that  $\mathcal{H}_U$  is actively causal to  $A$  given  $\mathcal{H}_V$ .
We say that  $\mathcal{H}_U$  has an active causal effect on  $\mathcal{F}$  given  $\mathcal{H}_V$ , or that  $\mathcal{H}_U$  is actively causal to  $\mathcal{F}$  given  $\mathcal{H}_V$ , if  $\mathcal{H}_U$  has an active causal effect on some  $A \in \mathcal{F}$ .
(iii) Otherwise, we say that  $\mathcal{H}_U$  has a dormant causal effect on  $A$  given  $\mathcal{H}_V$ , or that  $\mathcal{H}_U$  is dormantly causal to  $A$  given  $\mathcal{H}_V$ .
We say that  $\mathcal{H}_U$  has a dormant causal effect on  $\mathcal{F}$  given  $\mathcal{H}_V$ , or that  $\mathcal{H}_U$  is dormantly causal to  $\mathcal{F}$  given  $\mathcal{H}_V$ , if  $\mathcal{H}_U$  does not have an active causal effect on any event in  $\mathcal{F}$  given  $\mathcal{H}_V$  and there exists  $A \in \mathcal{F}$  on which  $\mathcal{H}_U$  has a dormant causal effect given  $\mathcal{H}_V$ .
Sometimes, we will say that  $\mathcal{H}_U$  has a causal effect on  $A$  given  $\mathcal{H}_V$  to mean that  $\mathcal{H}_U$  has either an active or a dormant causal effect on  $A$  given  $\mathcal{H}_V$ .
The intuition is as follows. For any fixed  $S \in \mathcal{P}(T)$  and any fixed event  $A \in \mathcal{H}$ , consider the function  $\omega_{S \cup V} \mapsto K_{S \cup V}((\omega_{(S \cup V) \setminus (U \setminus V)}, \omega_{S \cap (U \setminus V)}), A)$ . If  $\mathcal{H}_U$  has no causal effect on  $A$  given  $\mathcal{H}_V$ , then it means that the causal kernel does not depend on the  $\omega_{S \cap (U \setminus V)}$  component of  $\omega_{S \cup V}$ ; in other words,  $\mathcal{H}_U$  only has an influence on  $A$  through its  $V$  component.
We collect some important special cases in the following remark.
Remark B.5. (a) Letting  $V = U$ , we always have  $K_{S\cup U}(\omega ,A) = K_{(S\cup U)\backslash (U\backslash U)}(\omega ,A) = K_{S\cup U}(\omega ,A)$  for all  $\omega \in \Omega$  and  $A\in \mathcal{H}$ , which means that  $\mathcal{H}_U$  has no causal effect on any event  $A\in \mathcal{H}$  given itself.
(b) If  $\mathcal{H}_U$  has no causal effect on  $A$  given  $\mathcal{H}_V$ , then letting  $U = S$  in Definition B.4(i), we see that, for all  $\omega \in \Omega$ ,
$$
K _ {U \cup V} (\omega , A) = K _ {V} (\omega , A).
$$
In particular, this means that  $\mathcal{H}_U$  cannot have both no causal effect and active causal effect on  $A$  given  $\mathcal{H}_V$ .
(c) The case  $V = \emptyset$  reduces Definition B.4 to Definition B.1, i.e.  $\mathcal{H}_U$  having no causal effect in the sense of Definition B.1 is the same as  $\mathcal{H}_U$  having no causal effect given  $\{\emptyset, \Omega\}$  in the sense of Definition B.4, etc.
(d) It is possible for  $\mathcal{H}_U$  to be causal to an event  $A$ , and for there to exist  $V \in \mathcal{P}(T)$  such that  $\mathcal{H}_U$  has no causal effect on  $A$  given  $\mathcal{H}_V$ . However, if  $\mathcal{H}_U$  has no causal effect on  $A$ , then for any  $V \in \mathcal{P}(T)$ ,  $\mathcal{H}_U$  has no causal effect on  $A$  given  $\mathcal{H}_V$ . To see this, note that Remark B.2(e) tells us that  $U \setminus V$  also does not have any causal effect on  $A$ . Then given any  $S \in \mathcal{P}(T)$ ,
$$
K _ {S \cup V} (\omega , A) = K _ {(S \cup V) \backslash (U \backslash V)} (\omega , A),
$$
applying Definition B.1(i) to  $S \cup V$ . Since  $S \in \mathcal{P}(T)$  was arbitrary,  $\mathcal{H}_U$  has no causal effect on  $A$  given  $\mathcal{H}_V$ .
C Interventions
In this section, we provide a few more definitions and results related to the notion of interventions, introduced in Definition 2.3.
First, we make a few remarks on how the intervention causal kernels  $K_{S}^{\mathrm{do}(U,\mathbb{Q},\mathbb{L})}$  behave in some special cases, depending on the relationship between  $U$  and  $S$ .
Remark C.1. (a) For  $S \in \mathcal{P}(T)$  with  $U \subseteq S$ , we have, for all  $\omega \in \Omega$  and all  $A \in \mathcal{H}$ ,
$$
\begin{array}{l} K _ {S} ^ {\mathrm {d o} (U, \mathbb {Q}, \mathbb {L})} (\omega , A) = \int L _ {U} \left(\omega_ {U}, d \omega_ {U} ^ {\prime}\right) K _ {S} \left(\left(\omega_ {S \backslash U}, \omega_ {U} ^ {\prime}\right), A\right) \\ = \int \delta_ {\omega_ {U}} \left(d \omega_ {U} ^ {\prime}\right) K _ {S} \left(\left(\omega_ {S \backslash U}, \omega_ {U} ^ {\prime}\right), A\right) \quad \text {b y} \\ = K _ {S} \left(\left(\omega_ {S \backslash U}, \omega_ {U}\right), A\right) \\ = K _ {S} (\omega , A). \\ \end{array}
$$
This means that, after an intervention on  $\mathcal{H}_U$ , subsequent interventions on  $\mathcal{H}_S$  with  $\mathcal{H}_U \subseteq \mathcal{H}_S$  simply overwrite the original intervention. Note that this is reminiscent of the "partial ordering on the set of interventions" in [49], but in our setting, this is given by the partial ordering induced by the inclusion structure of sub- $\sigma$ -algebras of  $\mathcal{H}$ .
(b) For  $S\in \mathcal{P}(T)$  with  $S\subseteq U$
$$
K _ {S} ^ {\mathrm {d o} (U, \mathbb {Q}, \mathbb {L})} (\omega , A) = \int L _ {S} \left(\omega_ {S}, d \omega_ {U} ^ {\prime}\right) K _ {U} \left(\omega_ {U} ^ {\prime}, A\right)
$$
for all  $\omega \in \Omega$  and  $A\in \mathcal{H}$ , i.e.  $K_S^{\mathrm{do}(U,\mathbb{Q},\mathbb{L})}$  is a product of the two kernels  $K_{U}$  and  $L_{S}$  [14, p.39]; in particular,  $K_S^{\mathrm{do}(U,\mathbb{Q},\mathbb{L})}(\omega ,A) = L_S(\omega ,A)$  for all  $A\in \mathcal{H}_U$ .
(c) For  $S\in \mathcal{P}(T)$  with  $S\cap U = \emptyset$
$$
\begin{array}{l} K _ {S} ^ {\mathrm {d o} (U, \mathbb {Q}, \mathbb {L})} (\omega , A) = \int L _ {\emptyset} \left(\omega_ {\emptyset}, d \omega_ {U} ^ {\prime}\right) K _ {S \cup U} \left(\left(\omega_ {S}, \omega_ {U} ^ {\prime}\right), A\right) \\ = \int \mathbb {Q} \left(d \omega_ {U} ^ {\prime}\right) K _ {S \cup U} \left(\left(\omega_ {S}, \omega_ {U} ^ {\prime}\right), A\right) \quad \text {b y} \\ \end{array}
$$
for all  $\omega \in \Omega$  and  $A \in \mathcal{H}$ , i.e. the effect of intervening on  $\mathcal{H}_U$  with  $\mathbb{Q}$  then  $\mathcal{H}_S$  is the same as intervening on  $\mathcal{H}_{U \cup S}$  with a product measure of  $\mathbb{Q}$  on  $\mathcal{H}_U$  and whatever measure we place on  $\mathcal{H}_S$ .
We give it a name for the special case in which the internal causal kernels are all trivial (see Definition B.3).
Definition C.2. 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  $(\Omega, \mathcal{H}_U)$ . A hard intervention on  $\mathcal{H}_U$  via  $\mathbb{Q}$  is a new causal space  $(\Omega, \mathcal{H}, \mathbb{P}^{\mathrm{do}(U,\mathbb{Q})}, \mathbb{K}^{\mathrm{do}(U,\mathbb{Q},\mathrm{hard})})$ , where the intervention measure  $\mathbb{P}^{\mathrm{do}(U,\mathbb{Q})}$  is a probability measure  $(\Omega, \mathcal{H})$  defined in the same way as in Definition 2.3, and the intervention causal mechanism  $\mathbb{K}^{\mathrm{do}(U,\mathbb{Q},\mathrm{hard})} = \{K_S^{\mathrm{do}(U,\mathbb{Q},\mathrm{hard})}: S \in \mathcal{P}(T)\}$  consists of causal kernels that are obtained from the intervention causal kernels in Definition 2.3 in which  $L_{S \cap U}$  is a trivial causal kernel, i.e., one that has no causal effect on  $\mathcal{H}_{U \setminus S}$ .
From the discussion following Definition B.3, we have that, for  $A \in \mathcal{H}_{S \cap U}$  and  $B \in \mathcal{H}_{U \setminus S}$ ,  $L_{S \cap U}(\omega, A \times B) = 1_A(\omega_{S \cap U})\mathbb{Q}(B)$ .
The next result gives an explicit expression for the causal kernels obtained after a hard intervention.
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  $U \in \mathbb{P}(T)$  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
$$
K _ {S} ^ {\mathrm {d o} (U, \mathbb {Q}, \mathrm {h a r d})} (\omega , A) = K _ {S} ^ {\mathrm {d o} (U, \mathbb {Q}, \mathrm {h a r d})} (\omega_ {S}, A) = \int \mathbb {Q} (d \omega_ {U \setminus S} ^ {\prime}) K _ {S \cup U} ((\omega_ {S}, \omega_ {U \setminus S} ^ {\prime}), A).
$$
Intuitively, hard interventions do not encode any internal causal relationships within  $\mathcal{H}_U$ , so after we subsequently intervene on  $\mathcal{H}_S$ , the measure  $\mathbb{Q}$  that we originally imposed on  $\mathcal{H}_U$  remains on  $\mathcal{H}_{U\backslash S}$ .
The following lemma contains a couple of results about particular sub- $\sigma$ -algebras having no causal effects on particular events in the intervention causal space, regardless of the measure and causal mechanism that was used for the intervention.
Lemma C.4. 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  $U \in \mathcal{P}(T)$ ,  $\mathbb{Q}$  a probability measure on  $(\Omega, \mathcal{H}_U)$  and  $\mathbb{L} = \{L_V : V \in \mathcal{P}(U)\}$  a causal mechanism on  $(\Omega, \mathcal{H}_U, \mathbb{Q})$ . Suppose we intervene on  $\mathcal{H}_U$  via  $(\mathbb{Q}, \mathbb{L})$ .
(i) For  $A \in \mathcal{H}_U$  and  $V \in \mathcal{P}(T)$  with  $V \cap U = \emptyset$ ,  $\mathcal{H}_V$  has no causal effect on  $A$  (c.f. Definition B.1(i)) in the intervention causal space  $(\Omega, \mathcal{H}, \mathbb{P}^{\mathrm{do}(U,\mathbb{Q})}, \mathbb{K}^{\mathrm{do}(U,\mathbb{Q},\mathbb{L})})$ , i.e. events in the  $\sigma$ -algebra  $\mathcal{H}_U$  on which intervention took place are not causally affected by  $\sigma$ -algebras outside  $\mathcal{H}_U$ .
(ii) Again, let  $V \in \mathcal{P}(T)$  with  $V \cap U = \emptyset$ , and also let  $A \in \mathcal{H}$  be any event. If, in the original causal space,  $\mathcal{H}_V$  had no causal effect on  $A$ , then in the intervention causal space,  $\mathcal{H}_V$  has no causal effect on  $A$  either.
(iii) Now let  $V \in \mathcal{P}(T)$ ,  $A \in \mathcal{H}$  any event and suppose that the intervention on  $\mathcal{H}_U$  via  $\mathbb{Q}$  is hard. Then if  $\mathcal{H}_V$  had no causal effect on  $A$  in the original causal space, then  $\mathcal{H}_V$  has no causal effect on  $A$  in the intervention causal space either.
Lemma C.4(ii) and (iii) tell us that, if  $\mathcal{H}_V$  had no causal effect on  $A$  in the original causal space, then by intervening on  $\mathcal{H}_U$  with  $V\cap U = \emptyset$  or by any hard intervention, we cannot create a causal effect from  $\mathcal{H}_v$  on  $A$ . However, by intervening on a sub- $\sigma$ -algebra that contains both  $\mathcal{H}_V$  and (a part of)  $A$ , and manipulating the internal causal mechanism  $\mathbb{L}$  appropriately, it is clear that we can create a causal effect from  $\mathcal{H}_V$ .
The next result tells us that if a sub- $\sigma$ -algebra  $\mathcal{H}_U$  has a dormant causal effect on an event  $A$ , then there is a sub- $\sigma$ -algebra of  $\mathcal{H}_U$  and a hard intervention after which that sub- $\sigma$ -algebra has an active causal effect on  $A$ .
Lemma C.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, and  $U \in \mathcal{P}(T)$ . For an event  $A \in \mathcal{H}$ , if  $\mathcal{H}_U$  has a dormant causal effect on  $A$  in the original causal space, then there exists a hard intervention and a subset  $V \subseteq U$  such that in the intervention causal space,  $\mathcal{H}_V$  has an active causal effect on  $A$ .
The next result is about what happens to a causal effect of a sub- $\sigma$ -algebra that has no causal effect on an event conditioned on another sub- $\sigma$ -algebra, after intervening on that sub- $\sigma$ -algebra.
Lemma C.6. 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  $U, V \in \mathbb{P}(T)$ . For an event  $A \in \mathcal{H}$ , suppose that  $\mathcal{H}_U$  has no causal effect on  $A$  given  $\mathcal{H}_V$  (see Definition B.4). Then after an intervention on  $\mathcal{H}_V$  via any  $(\mathbb{Q}, \mathbb{L})$ ,  $\mathcal{H}_{U \setminus V}$  has no causal effect on  $A$ .
The next result shows that, under a hard intervention, a time-respecting causal mechanism stays time-respecting.
Theorem C.7. 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, where the index set  $T$  can be written as  $T = W \times \tilde{T}$ , with  $W$  representing time and  $\mathbb{K}$  respecting time. Take any  $U \in \mathcal{P}(T)$  and any probability measure  $\mathbb{Q}$  on  $\mathcal{H}_U$ . Then the intervention causal mechanism  $\mathbb{K}^{\mathrm{do}(U, \mathbb{Q}, \mathrm{hard})}$  also respects time.
D Sources
In causal spaces, the observational distribution  $\mathbb{P}$  and the causal mechanism  $\mathbb{K}$  are completely decoupled. In Section 3.1, we give a detailed argument as to why this is desirable, but of course, there is no doubt that the special case in which the causal kernels coincide with conditional measures with respect to  $\mathbb{P}$  is worth studying. To that end, we introduce the notion of sources.
Definition D.1. 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,  $U \in \mathcal{P}(T)$ ,  $A \in \mathcal{H}$  an event and  $\mathcal{F}$  a sub- $\sigma$ -algebra of  $\mathcal{H}$ . We say that  $\mathcal{H}_U$  is a (local) source of  $A$  if  $K_U(\cdot, A)$  is a version of the conditional probability  $\mathbb{P}_{\mathcal{H}_U}(A)$ . We say that  $\mathcal{H}_U$  is a (local) source of  $\mathcal{F}$  if  $\mathcal{H}_U$  is a source of all  $A \in \mathcal{F}$ . We say that  $\mathcal{H}_U$  is a global source of the causal space if  $\mathcal{H}_U$  is a source of all  $A \in \mathcal{H}$ .
Clearly, source  $\sigma$ -algebras are not unique (whether local or global). It is easy to see that  $\mathcal{H}_{\emptyset} = \{\emptyset, \Omega\}$  and  $\mathcal{H} = \mathcal{H}_T = \otimes_{t \in T} \mathcal{E}_t$  are global sources, and axiom (ii) of Definition 2.2 implies that any  $\mathcal{H}_S$  is a local source of any of its sub- $\sigma$ -algebras, including itself, since, for any  $A \in \mathcal{H}_U$ ,  $\mathbb{P}_{\mathcal{H}_U}(A) = 1_A$ . Also, a sub- $\sigma$ -algebra of a source is not necessarily a source, nor is a  $\sigma$ -algebra that contains a source necessarily a source (whether local or global). In Example 2.5 above, altitude is a source of temperature (and hence a global source), since the causal kernel corresponding to temperature coincides with the conditional measure given altitude, but temperature is not a source of altitude.
When we intervene on  $\mathcal{H}_U$  (via any  $(\mathbb{Q},\mathbb{L}))$ ,  $\mathcal{H}_U$  becomes a global source. This precisely coincides with the "gold standard" that is randomised control trials in causal inference, i.e. the idea that, if we are able to intervene on  $\mathcal{H}_U$ , then the causal effect of  $\mathcal{H}_U$  on any event can be obtained by first intervening on  $\mathcal{H}_U$ , then considering the conditional distribution on  $\mathcal{H}_U$ . Next is a theorem showing that when one intervenes on  $\mathcal{H}_U$ , then  $\mathcal{H}_U$  becomes a source.
Theorem D.2. Suppose we have a causal space  $(\Omega, \mathcal{H}, \mathbb{P}, \mathbb{K}) = (\times_{t \in T} E_t, \otimes_{t \in T} \mathcal{E}_t, \mathbb{P}, \mathbb{K})$ , and let  $U \in \mathcal{P}(T)$ .