## Definition The definitions of hypermeasurability are very similar to the definitions of [[strong|strongness]], mainly because hypermeasurability is a generalised version of strongness. The intuition behind each definition is also very similar to that of the matching definitions of strongness. ### Elementary Embedding Characterisation A cardinal $\kappa$ is **$x$-hypermeasurable** for a set $x$ iff it is the critical point of some elementary embedding $j:V\rightarrow M$ for some transitive class $M$ such that $x\in M$. A cardinal $\kappa$ is **$\lambda$-hypermeasurable** iff it is $H_\lambda$-hypermeasurable (where $H_\lambda$ is the set of all sets of [hereditary cardinality](Hereditary_Cardinality "Hereditary Cardinality") less than $\lambda$). Note that a cardinal is $\gamma$-strong iff it is $x$-hypermeasurable for every $x\in V_\gamma$ (iff it is $V_\gamma$-hypermeasurable as well) and a cardinal is strong iff it is $x$-hypermeasurable for every $x$. # Facts - A cardinal $\kappa$ is [measurable](Measurable.md "Measurable") if and only if it is $\kappa^+$-hypermeasurable, since $\mathcal{P}(\kappa)\subset M$ for any $j:V\to M$ with critical point $\kappa$. - If there is an $x$-hypermeasurable cardinal, then $V\neq L[x]$. {% cite Jech2003 %} - A cardinal $\kappa$ is $\gamma$-strong if and only if $\kappa$ is $\beth_\gamma$-hypermeasurable, by definition. - In particular, $\kappa$ is $\mathcal{P}^2(\kappa)$-hypermeasurable if and only if it is $\kappa+2$-strong. This hypothesis appears in many theorems.