We know that if ‎$‎‎\kappa‎‎$ is a‎ ‎measurable ‎cardinal ‎and ‎‎$‎‎\mu$ be a‎ ‎two-valued ‎non-trivial‎‎$‎‎‎\kappa‎$-additive ‎measure ‎on ‎it ‎the‎n the corresponding inner model produced by Mostowski collapse of Scott's ultraproduct (‎$‎‎M\_{‎\kappa‎,\mu}$‎) is dependent on both ‎$‎‎‎\kappa‎$ ‎and ‎‎$‎‎\mu$. ‎So by ‎changing ‎measures ‎‎on a certain measurable cardinal we ‎can ‎produce ‎many different ‎inner ‎models. Even by transitivity if two such inner models be isomorphic then they are equal. Now a question is:


"Is it possible to have a measurable cardinal which its inner models be independent from its measures?"


Precisely: 


**Definition (1):** A ‎"unique ‎measurable ‎cardinal" ‎is a‎ ‎measurable ‎cardinal which for all two-valued ‎non-trivial ‎‎$‎‎‎\kappa‎$-additive ‎measures like $\mu$ and $\mu'$ ‎on ‎it we have $M\_{\kappa , \mu}=M\_{\kappa , \mu'}$. We show this unique inner model by $M\_{\kappa}$. 


**Question (1):** Is the following statement true?


$Con(ZFC+ \text{there exists a measurable cardinal})\Longrightarrow$
$Con(ZFC+ \text{there exists a unique measurable cardianl})$
‎


‎Now ‎the ‎main ‎question ‎is ‎about ‎the ‎behavio‎r of the "well defined" function ‎$‎\kappa ‎‎\mapsto ‎M\_{‎\kappa‎}‎‎‎‎$:‎
‎


**Definition (2):** Define:‎
‎


The collection of all unique measurable cardinals: ‎
‎


$‎‎‎UM:=‎\lbrace ‎‎‎\kappa‎~|~‎\kappa~‎\text{is a unique measurable cardinal‎}‎\rbrace‎$ ‎‎
‎


The ‎(informal) ‎collection ‎of ‎all ‎inner ‎models ‎of ‎unique ‎measurable ‎cardinals: ‎
‎


‎‎$‎‎IUM:=‎\lbrace ‎‎‎M\_{\kappa}‎~|~‎\kappa ‎\in UM ‎\rbrace‎‎‎$‎
‎


**Question (2):** ‎Is ‎the ‎function‎ ‎$‎\kappa ‎‎\mapsto ‎M\_{‎\kappa‎}‎$ ‎from ‎‎$‎‎UM$ ‎to ‎‎$‎‎IUM$ (strictly) ‎‎increasing? ‎In ‎the ‎other ‎words which one of the following statements are true?‎
‎


‎‎$‎‎(1)~‎\forall ‎‎\kappa‎,‎\lambda ‎\in UM~~~~~(‎\kappa < ‎‎‎\lambda ‎‎\longrightarrow ‎M\_{‎\kappa‎}\subseteq M\_{‎\lambda‎}‎‎‎‎‎)‎‎$‎
‎


$‎‎(2)~‎\forall ‎‎\kappa‎,‎\lambda ‎\in UM~~~~~(‎\kappa < ‎‎‎\lambda ‎‎\longrightarrow ‎M\_{‎\kappa‎}\subsetneq M\_{‎\lambda‎}‎‎‎‎‎)‎‎$‎
‎‎ 



Do inner models of unique measurable cardinals have a regular behavior? (Edited and Revised Version)
Because Question 1 asked about arbitrary two-valued, $\kappa$-additive, non-trivial measures on $\kappa$, not only normal ones, the answer is negative. (If one considers only normal measures, then the answer becomes positive, as explained in a comment by Asaf Karagila.) If $U$ is a measure on $\kappa$ (by which I mean two-valued, $\kappa$-additive, and non-trivial throughout this answer), then there is a measure $U\otimes U$ on $\kappa\times\kappa$ defined as 
$$
\{X\subseteq\kappa\times\kappa:\{\alpha<\kappa:\{\beta<\kappa:(\alpha,\beta)\in X\}\in U\}\in U\}.
$$
Applying a bijection from $\kappa\times\kappa$ to $\kappa$, we get a measure $U^2$ on $\kappa$. The ultrapower (by which I always mean its transitive collapse in this answer) of the universe with respect to this $U^1$ (or with respect to the isomorphic $U\otimes U$) is the two-step iterated ultrapower of the universe with respect to $U$, so it is a proper submodel of the (one-step) ultrapower of the universe with respect to $U$. 


