Proof. For the backwards implication, since $\kappa $ is inaccessible in $\operatorname {\mathrm {HOD}}$ and $\eta < \kappa $ , $\left (2^{\eta }\right )^{\operatorname {\mathrm {HOD}}} < \kappa $ .

Let $\langle T_{\alpha } \mid \alpha < (\eta ^+)^{\operatorname {\mathrm {HOD}}}\rangle \in \operatorname {\mathrm {HOD}}$ be a decomposition of S into stationary sets. By the assumption, for each $\alpha $ there is a measure $U_{\kappa ,i}$ in $\operatorname {\mathrm {HOD}}$ such that $T_{\alpha } \in U_{\kappa ,i}$ . Since the sets $T_{\alpha }$ are pairwise disjoint, it is impossible for $\alpha \neq \beta $ to belong to the same $U_{\kappa ,i}$ . Thus, we obtain an injective function from $(\eta ^+)^{\operatorname {\mathrm {HOD}}}$ to $\eta $ in $\operatorname {\mathrm {HOD}}$ – a contradiction.

Let us assume now that $\kappa $ is strongly measurable with respect to $S \in \operatorname {\mathrm {HOD}}$ . In particular, $\kappa $ is inaccessible in $\operatorname {\mathrm {HOD}}$ . Let $\mathcal {S} \in \operatorname {\mathrm {HOD}}$ be a maximal collection of pairwise disjoint stationary subsets of S, in $\operatorname {\mathrm {HOD}}$ , such that for all $T \in \mathcal S$ , the club filter restricted to T is an ultrafilter in $\operatorname {\mathrm {HOD}}$ . Let us denote this ultrafilter by $U_T$ . Since this collection is a partition of S into stationary sets, $|\mathcal S| < \kappa $ .

If $\bigcap \{U_T \mid T \in \mathcal S\}$ is not the club filter restricted to S in $\operatorname {\mathrm {HOD}}$ , then it contains a set $S \setminus S'$ , where $S' \subseteq S$ stationary, $S' \in \operatorname {\mathrm {HOD}}$ . In particular, $S' \notin U_T$ for all $T \in \mathcal S$ , so $S' \cap T$ is nonstationary for all $T \in \mathcal S$ , and $S" = S' \setminus \bigcup _{T \in \mathcal S} (S' \cap T)$ is a stationary subset of S, disjoint from all members of $\mathcal S$ . Since $\kappa $ is strongly measurable with respect to S, it is also strongly measurable with respect to $S"$ . Thus, there is some $T' \subseteq S"$ stationary such that the club filter restricted to $T'$ is an ultrafilter. But this contradicts the maximality of $\mathcal {S}$ .

Proof. Fix any $f\colon H_{\theta }^{<\omega } \to H_{\theta }$ in $V[H]$ . We would like to show that there exists some $M \subseteq H_{\theta }$ which is closed under f and satisfies the conditions in the statement of the lemma.

Fix in V a name for f and let $f' \colon \operatorname {\mathrm {Coll}}(\omega , <\kappa ) \times (H_{\theta }^V)^{<\omega } \to H_{\theta }^V$ be a function that sends $(p, x) \in \operatorname {\mathrm {Coll}}(\omega ,<\kappa )\times (H_{\theta })^{<\omega }$ to y if . Note that x is a finite sequence of names.

By the definition of $f'$ , if $M' \prec H_{\theta }^V$ is closed under $f'$ , $M' \cap \kappa \in \kappa $ , and $\kappa , U \in M'$ , then $M' = M \cap H_{\theta }^V$ for some $M \subseteq H_{\theta }$ which is closed under f. Indeed, we may take $M = M'[H \cap M]$ . By the chain condition of $\operatorname {\mathrm {Coll}}(\omega ,<\kappa )$ , every name for a ground model object that belongs to $M'$ can be refined to a nice name which is contained in $M'$ .

Since $U \subseteq H_{\theta }^V$ , it is therefore sufficient to prove that there exists some $M' \subseteq H_{\theta }^V$ which is closed under $f'$ and satisfies $|M'| < \kappa $ and $\sup (M' \cap \kappa ) \in A$ for all $A \in M' \cap U$ .

Working in V, take an elementary substructure $N \prec H_{\theta }^V$ satisfying $f'[N] \subseteq N$ , $N^{<\kappa } \subseteq N$ , $|N| = \kappa $ , $\kappa \in N$ . Let $j \colon V \to W \cong \mathrm {Ult}(V,U)$ be the ultrapower embedding induced by U. Consider the structure $\tilde {M}' = j" N \prec j(H_{\theta }^V)$ . $\tilde {M}' \in W$ is closed under $j(f')$ , and $\tilde {M}' \cap j(\kappa ) \in j(\kappa )$ . It follows that $0_{j(\operatorname {\mathrm {Coll}}(\omega ,<\kappa ))}$ forces $\tilde {M}'$ to be closed under . Finally, for every $A \in \tilde {M}' \cap j(U)$ , $A = j(\bar {A})$ for some $\bar {A}\in U$ and therefore, $\kappa \in A$ .

So $\tilde {M}$ satisfies the conclusion of the lemma in W, and it is closed under $j(f')$ . By the elementarity of j, there is $M' \in V$ satisfying the conclusion of the lemma and closed under $f'$ . Thus, $M'[H \cap M'] = M$ satisfies the requirements of the lemma.

Proof. Since $\kappa = \aleph _1$ in $V[H]$ , we need to check that the intersection of a countable family $\{ D_n \mid n<\omega \}$ of dense open subsets of $\mathbb {C}_{U}$ is dense. Pick some regular cardinal $\theta> \kappa ^+$ such that $\mathbb {C}_U, \{D_n \mid n<\omega \} \in H_{\theta }$ . By lemma 3.1, for every condition $x \in \mathbb {C}_{U}$ , there exists an elementary substructure $M \prec H_{\theta }$ of size $|M| = \aleph _0$ , with $x, \mathbb {P},\mathbb {C}_{U},\{D_n \mid n < \omega \} \in M$ and further satisfies that $\sup (M \cap \kappa ) \in A$ for every $A \in M \cap \kappa $ . We may also assume that $M = M'[H \cap M']$ for $M' \in V$ , $M' \prec H_{\theta }^V$ .

Denote $\sup (M \cap \kappa )$ by $\alpha $ and pick a cofinal sequence ${\langle } \alpha _n \mid n < \omega {\rangle }$ in $\alpha $ . We can construct an increasing sequence of extensions ${\langle } x_n \mid n<\omega {\rangle } \subseteq M$ of x, $x_n = {\langle } c_n,A_n{\rangle }$ such that $x_{n+1} \in D_n$ and $\max (c_n) \geq \alpha _n$ for every $n<\omega $ . Without loss of generality, we may assume that for every $A \in U \cap M$ , there is $n < \omega $ such that $A_n \subseteq A$ .

Since $x_n = {\langle } c_n,A_n{\rangle } \in M$ , then $\alpha \in A_n$ for all $n < \omega $ . It follows that $x^* = {\langle } \{ \alpha \} \cup (\bigcup _n c_n), \bigcap _n A_n{\rangle }$ is a condition in $\mathbb {C}_U$ , which is clearly an upper bound of ${\langle } x_n \mid n < \omega {\rangle }$ .Footnote ⁵ We conclude that there exists $x^*$ extending our given condition x such that $x^* \in \bigcap _n D_n$ .

Proof. Let $x_1 = {\langle } c_1,A_1{\rangle }$ , $x_2 = {\langle } c_2,A_2{\rangle }$ be two conditions of $\mathbb {C}_{U}$ . Take $\nu \in A_1 \cap A_2$ above $\max (c_1),\max (c_2)$ and consider the extensions $y_1 = {\langle } c_1 \cup \{\nu \}, (A_1 \cap A_2) \setminus (\nu +1){\rangle }$ , $y_2 = \langle c_2 \cup \{\nu \}, (A_1 \cap A_2) \setminus (\nu +1){\rangle }$ of $x_1$ Define a cone isomorphism $\sigma \colon \mathbb {C}_U/y_1 \to \mathbb {C}_U/y_2$ by

$$\begin{align*}\sigma( {\langle} c, A{\rangle}) = {\langle} c_2 \cup (c\setminus \nu), A{\rangle} \end{align*}$$

$\sigma $ is clearly an order preserving map onto $\mathbb {C}_U/y_1$ and has an order preserving inverse which is given by

$$\begin{align*}\sigma^{-1}( {\langle} c, A{\rangle}) = {\langle} c_1 \cup (c\setminus \nu), A{\rangle} \end{align*}$$

Proof. By Lemma 8.3, $\operatorname {\mathrm {Coll}}(\omega ,<\kappa ) * \mathbb {C}_U$ is cone homogeneous, and therefore, $\operatorname {\mathrm {HOD}}^{V[H \ast C]} \subseteq V$ . It is clear from the definition of $\mathbb {C}_U$ that for every subset $S \subseteq \kappa $ in V, S is stationary in $V[H \ast C]$ if and only if $S \in U$ . It follows that the closed unbounded filter on $\kappa = \aleph _1^{V[H \ast C]}$ in $V[H \ast C]$ is a $\operatorname {\mathrm {HOD}}$ -ultrafilter. Therefore $V[H \ast C]\models \kappa $ is strongly measurable.

Proof. Suppose that ${\langle } A_{\nu } \mid \nu < \beta {\rangle } \in V[G\ast H]$ is a sequence of $\beta < \kappa $ many sets of $\mathcal {F}_{\kappa }$ . We would like to show that $\bigcap _{\nu < \beta } A_{\nu }$ belongs to $\mathcal {F}_{\kappa }$ . We may assume that $A_{\nu } \in \bigcap _{i \leq \lambda } U_{\kappa _i}$ for all $\nu < \beta $ .

In order to prove the claim, we move from $V[G\ast H]$ to $V[G]$ , and then to V. Working in $V[G]$ , let be a $\operatorname {\mathrm {Coll}}(\lambda ,<\kappa )$ -name for the V-set in $\bigcap _{i\leq \lambda } U_{\kappa ,i}$ . Since $\operatorname {\mathrm {Coll}}(\lambda ,<\kappa )$ satisfies $\kappa $ -c.c., there exists a family of V-sets $X_{\nu } \subseteq \bigcap _{i\leq \lambda } U_{\kappa ,i}$ , $X_{\nu } \in V[G]$ , of size $<\kappa $ such that . Fix in V a $\mathbb {P}$ -name for each $X_{\nu }$ .

Let $p \in G$ be a condition forcing the above. Moving back to V, the fusion lemma for nonstationary support iteration of Prikry type forcings [Reference Ben-Neria and Unger5, Lemma 3.6] guarantees that there exists some $q \in G$ and a sequence of sets ${\langle } Y_{\nu } \mid \nu < \beta {\rangle }$ in V, so that for each $\nu < \beta $ , $Y_{\nu } \subseteq \bigcap _{i \leq \lambda }U_{\kappa ,i}$ has size $|Y_{\nu }| < \kappa $ , and . For each $\nu $ , let $A^{\prime }_{\nu } = \bigcap Y_{\nu }$ , and $A' = \bigcap _{\nu < \beta } A^{\prime }_{\nu }$ . Since $U_{\kappa ,i}$ is $\kappa $ -complete for all $i \leq \lambda $ , we have in $V[G\ast H]$ that $A' \in \mathcal {F}_{\kappa }$ and $A' \subseteq \bigcap _{\nu < \beta } A_{\nu }$ .

Proof. Fix a function $f\colon [H_{\theta }]^{<\omega } \to H_{\theta }$ in $V[G\ast H]$ . Back in the ground model V, let be a $\mathbb {P} * \operatorname {\mathrm {Coll}}(\lambda ,<\kappa )$ -name for f. Since $\operatorname {\mathrm {Coll}}(\lambda ,<\kappa )$ is $\kappa $ -c.c., there exists a $\mathbb {P}$ -name function such that is forced to be a member of for every $x \in H_{\theta }^V$ .

Let us consider our ability to approximate in V. Let $N \prec H_{\theta }^V$ be an elementary substructure of size $\kappa $ with $N^{<\kappa } \subseteq N$ and .

Let $j_{\tau }\colon V \to M_{\tau }$ be the ultrapower embedding by $U_{\kappa ,\tau }$ and $M' = j_{\tau }" N \prec j_{\tau }(H_{\theta }^V)$ , $M' \in M_{\tau }$ .

We now return to prove the statement of the lemma. It is sufficient to prove that in $V[G]$ there exists some $M' \subseteq H_{\theta }^V$ which is closed under F and satisfies requirements (i)–(iii). Let $p \in \mathbb {P}$ be a condition. By a standard density argument, there are $N \prec H_{\theta }^V$ and $p^* \in G$ which is N-generic, with $p^* \geq ^* p$ .Footnote ⁶ By Claim 4.7, $j_{\tau }(p^*)$ forces that $M' = j_{\tau }\text {"} N$ is closed under $j_{\tau }(F)$ . It is now clear that $M'$ satisfies condition (ii) in the ultrapower, as $o^{j_{\tau }(\mathcal {U})}(\kappa ) = \tau $ and $M' \cap j_{\tau }(\kappa ) = \kappa $ . Condition (i) holds as well, since $j(\mathbb {P}_{\kappa }) / \mathbb {P}_{\kappa }$ does not introduce new ${<}\tau $ -sequences to $j_{\tau } \text {"} N$ . Therefore, it remains to verify that $j_{\tau }(p^*)$ forces $M'$ to satisfy condition (iii). For every $A \in M' \cap j_{\tau }(\mathcal {F}_{\kappa })$ , there is some $B \in \mathcal {F}_{\kappa }$ such that $A = j_{\tau }(B)$ . In particular, $A \cap \kappa = B \in \mathcal {F}_{\kappa }$ and $\kappa \in A$ . Since $\mathcal {F}_{\kappa } \subseteq \bigcap _{i \leq \tau } U_{\kappa ,i}$ (which is $\mathcal {F}_{\kappa }^{M_{\tau }}$ ), it follows form remark 4.1 that for every generic filter $G^* \subseteq j_{\tau }(\mathbb {P})$ over $M_{\tau }$ , if $b_{\kappa }$ is the $G^*$ -induced $\mathbb {Q}_{\kappa }^{\tau }$ cofinal generic sequence, then it is almost contained in $B = A \cap \kappa $ .

Proof. Since $\kappa = \lambda ^+$ in $V[G\ast H]$ , we need to check that the intersection of every set $\{ D_i \mid i < \lambda \}$ of $\lambda $ -many dense open subsets of $\mathbb {C}_{\mathcal {F}_{\kappa }}$ is dense. Pick some regular cardinal $\theta> \kappa ^+$ such that $\mathbb {P}, \mathbb {C}_{\mathcal {F}_{\kappa }}, \{D_i \mid i < \lambda \} \in H_{\theta }$ . By Lemma 4.5, for every condition $x \in \mathbb {C}_{\mathcal {F}}$ , there exists an elementary substructure $M \prec H_{\theta }$ of cardinality $< \kappa $ , with $x, \mathbb {P},\mathbb {C}_{\mathcal {F}_{\kappa }},\{D_i \mid i < \lambda \} \in M$ and which further satisfies (i) $M^{<\lambda } \subseteq M$ ; (ii) $\sup (M \cap \kappa ) = \alpha $ has $o^{\mathcal {U}}(\alpha ) = \lambda $ ; and (iii) $\alpha \in A$ and $b_{\alpha }$ is almost contained in A for every $A \in \mathcal {F}_{\kappa } \cap M$ .

Let ${\langle } \alpha _i \mid i < \lambda {\rangle }$ be an increasing enumeration of $b_{\alpha }$ . We construct by induction an increasing sequence of extensions ${\langle } x_j \mid j < \lambda {\rangle }$ of x, together with an increasing subsequence ${\langle } \alpha _{i_j} \mid j < \lambda {\rangle }$ of $b_{\alpha }$ such that $x_{j+1} \in D_j$ for every $j < \lambda $ , and $\{\alpha \} \cup \{ \alpha _{i_j} \mid j> j^*\} \subseteq A^{x_{j^*}}$ for all $j^* < \lambda $ . For notational simplicity, denote x by $x_{-1}$ . Given a condition $x_j \in M$ with a suitable $\alpha _j$ as above, we take $x_{j+1}\in D_{j+1}$ to be an extension of $x_j$ with $\max (c^{x_{j+1}})> \alpha _{i_j}$ . Since $A^{x_{j+1}} \in M \cap \mathcal {F}_{\kappa }$ , we can use (iii) and get that $\alpha \in A^{x_{j+1}}$ and there exist some $i'> i_j$ such that $\{ \alpha _i \mid i>i'\} \subseteq A^{x_{j+1}}$ . Take $i_{j+1} < \lambda $ to be the minimal such $i'> i_j$ . It remains to show that the construction goes through at limit stages $\delta \leq \lambda $ . Given ${\langle } x_j \mid j < \delta {\rangle }$ , we define $i_{\delta } = \sup _{j < \delta } \alpha _{i_j}$ . It is clear from our construction at successor steps that $\alpha _{i_{\delta }} = \sup _{j < \delta } \max (c^{x_j})$ and $\alpha _{i_{\delta }} \in A^{x_j}$ for every $j < \delta $ . It follows that the condition $x_{\delta } = {\langle } \{ \alpha _{\delta } \} \cup \bigcup _{j < \delta } c^{x_j}, \bigcap _{j<\delta } A^{x_j}{\rangle }$ satisfies the desirable conditions. Moreover, if $\delta < \lambda $ , then $x_{\delta } \in M$ since M is closed under $<\lambda $ -sequences.

Since the limit construction goes through at stage $\lambda $ as well (although not producing a condition in M), the limit condition $x_{\lambda }$ is an extension of x and belongs to $\bigcap _{j <\lambda } D_j$ .

Proof. Suppose $G(\mathbb {C}_{\mathcal {F}_{\kappa }}) \subseteq \mathbb {C}_{\mathcal {F}_{\kappa }}$ is a generic filter over $V[G\ast H]$ . We may identify $G(\mathbb {C}_{\mathcal {F}_{\kappa }})$ with its derived generic closed and unbounded set

$$\begin{align*}C = \bigcup \{ c \mid \exists A {\langle} c,A{\rangle} \in G(\mathbb{C}_{\mathcal{F}_{\kappa}})\}. \end{align*}$$

By a standard density argument, we have that for every set $X \subseteq \kappa $ in V, if $X \notin U_{\kappa ,\tau }$ for all $\tau \leq \lambda $ , then $|C \cap X| < \kappa $ .

We conclude that for $X \subseteq \kappa $ in V to be stationary in $V[G \ast H \ast C]$ , it must belong to $U_{\kappa ,\tau }$ for some $\tau \leq \lambda $ . It follows that if ${\langle } S_i \mid i <\eta {\rangle } \subseteq V$ is a partition of $\kappa $ into disjoint sets which are stationary in $V[G \ast H \ast C]$ , then $|\eta | \leq \lambda $ . Moreover, since $\kappa $ is inaccessible in V, we have $(2^{\eta })^V < \kappa $ .

Finally, we know that each poset $\mathbb {P}$ , $\operatorname {\mathrm {Coll}}(\lambda ,<\kappa )$ , and $\mathbb {C}_{\mathcal {F}_{\kappa }}$ is forced in turn to be cone homogeneous and clearly definable using parameters from the ground model. Therefore, $\mathbb {P}* \operatorname {\mathrm {Coll}}(\lambda ,<\kappa ) \ast \mathbb {C}_{\mathcal {F}_{\kappa }}$ is cone homogeneous, and therefore, $\operatorname {\mathrm {HOD}}^{V[G \ast H \ast C]} \subseteq V$ . The claim follows.

Question 4.11. Is it consistent that the club filter restricted to $S^{\lambda }_{\omega }$ is an ultrafilter in $\operatorname {\mathrm {HOD}}$ for a regular cardinal $\lambda> \aleph _2$ ?

Question 4.12. What is the consistency strength of $\omega _2$ being $\omega $ -strongly measurable in $\operatorname {\mathrm {HOD}}$ ?

Proof (Theorem 1.4).

To simplify our arguments, we work over a minimal Mitchell model $V = L[\mathcal {U}]$ with a coherent sequence of measures $\mathcal {U}$ witnessing the assumed large cardinal assumption. Therefore, $\theta $ is the least inaccessible cardinal in V for which $\{ o(\kappa ) \mid \kappa < \theta \}$ is unbounded in $\theta $ . We note that all normal measures in this model appear on the main sequence $\mathcal {U}$ , in particular, $o(\kappa ) = o^{\mathcal {U}}(\kappa )$ for all $\kappa $ . We also record here that by the Mitchell Covering Theorem and the fact $\theta $ is not measurable, there is no generic extension of $V = L[\mathcal {U}]$ which preserves the cardinals below $\theta $ and changes the cofinality of $\theta $ . Similarly, the Mitchell Covering Theorem guarantees that generic extensions of $V = L[\mathcal {U}]$ satisfy the Weak Covering Lemma with respect to V, which implies that successors of singular cardinals cannot be collapsed.

Let $\langle \kappa _{\alpha } \mid \alpha < \theta \rangle $ be an increasing sequence of cardinals below $\theta $ , which satisfies the following conditions:

1. 1. $\kappa _0 = \omega $ , $\kappa _1$ is the least measurable,

2. 2. for a limit ordinal $\alpha $ , $\kappa _{\alpha } = (\sup _{\beta < \alpha } \kappa _{\beta } )^+$ ,

3. 3. for a successor ordinal $\alpha + 1$ , let $\kappa _{\alpha + 1}$ be the least cardinal such that $o^{\mathcal {U^{\alpha }}}(\kappa _{\alpha + 1}) = \kappa _{\alpha } + 1$ , for the coherent sequence of measures $\mathcal {U}^{\alpha } = \mathcal {U}\restriction _{(\kappa _{\alpha },\kappa _{\alpha +1}]}$ . In particular, the first measure of the sequence $\mathcal {U}^{\alpha }$ has critical point $> \kappa _{\alpha }$ .

We define by induction on $\alpha < \theta $ a Magidor iteration $\mathbb {P} = {\langle } \mathbb {P}_{\alpha }, \mathbb {Q}_{\alpha } \mid \alpha < \theta {\rangle }$ of Prikry type forcings. Our description of the Magidor style iteration follows Gitik’s handbook chapter [Reference Gitik13]. We recall that conditions are sequences of the form $\langle q_{\alpha } \mid \alpha < \theta \rangle $ where only finitely many coordinates are not a direct extension of the weakest condition $0_{\mathbb {Q}_{\alpha }}$ . Let $\mathbb {Q}_{0}$ be $\operatorname {\mathrm {Coll}}(\omega , <\kappa _1) * \mathbb {C}^*_{\mathcal {F}_{\kappa _1}}$ , where $\mathcal {F}_{\kappa _1}$ is the filter generated from the normal measure on $\kappa _1$ . For $\alpha> 0$ , we define $\mathbb {Q}_{\alpha } = \mathbb {Q}[\mathcal {U}^{\alpha }]$ .

The coherent sequence $\mathcal {U}^{\alpha }$ from $L[\mathcal {U}]$ uniquely extends in a generic extension by $\mathbb {P}_{\alpha }$ and can therefore be used to force with $\mathbb {Q}[\mathcal {U}^{\alpha }]$ . This is because as $L[\mathcal {U}]$ satisfies the $\operatorname {\mathrm {GCH}}$ , we have that $|\mathbb {P}_{\alpha }| \leq \kappa _{\alpha }$ and all measures of $\mathcal {U}^{\alpha }$ are assumed to have critical points strictly above $\kappa _{\alpha }$ . It is clear from our definitions that $\mathbb {Q}_{\alpha }$ satisfies the Prikry Property, that its direct extension order is $\kappa _{\alpha }$ -closed and that $\mathbb {Q}_{\alpha }$ is forced to be cone homogeneous.

By the general theory of Magidor iteration of Prikry type posets, the iteration $\mathbb {P}_{\theta }/ \mathbb {P}_1$ also satisfies the Prikry Property. Moreover, for every $\alpha < \theta $ , $\mathbb {P}_{\theta } / \mathbb {P}_{\alpha }$ has the Prikry Property in the generic extension by $\mathbb {P}_{\alpha }$ , and its direct extension order is $\kappa _{\alpha }$ -closed (see [Reference Gitik13] for details).

Let $G_{\theta } \subseteq \mathbb {P}_{\theta }$ be a generic filter over V. We conclude that $\operatorname {\mathrm {HOD}}^{V[G_{\theta }]} \subseteq V$ .Footnote ⁸ Moreover, for each $\alpha < \kappa $ , the $\mathbb {Q}[\mathcal {U}^{\alpha }]$ generic filter induced by $G_{\theta }$ guarantees that $\kappa _{\alpha +1} = (\kappa _{\alpha }^+)^{V[G_{\theta }]}$ and that $\kappa _{\alpha }$ is strongly measurable in $\operatorname {\mathrm {HOD}}^{V[G_{\theta }]}$ . It follows that all successors of regular cardinals below $\theta $ in $V[G_{\theta }]$ are strongly measurable in $\operatorname {\mathrm {HOD}}$ . Since $\theta $ remains strongly inaccessible in $V[G_{\theta }]$ and all the relevant witnessing objects clearly belong to $V_{\theta }^{V[G_{\theta }]}$ , we conclude that in $V_{\theta }^{V[G_{\theta }]}$ , all successors of regular cardinals are strongly measurable.