Let $R$ denote your favorite hyperreal ordered field and let $\delta\in R$ be a positive infinitesimal. Let $\mathcal{E}$ denote the set of all (standard) finite Boolean subalgebras of $\mathcal{P}([0,1))$. For every $A\in\mathcal{E}$, let $\lambda_A(I)$ be the (exact) length of $I$ for all half-open intervals $I\in A$; for all open or closed intervals $I\in A$, respectively subtract or add $\delta$ to the length of $I$ to define $\lambda_A(I)$; let $\lambda_A(S)=\delta$ for all singletons $S\in A$. Extend $\lambda_A$ to a probability measure $\mu_A$ on $A$.
(Specifically, for each minimal interval finite union of intervals $I\in F\in A$that is not an atom , let the connected components of $A$, partition F$be$I$[a_0,b_0),\ldots,[a_k,b_k)$ with $p$ elements of $\{a_i,b_i:i\leq k\}$ added and $q$ removed. Partition $F$ into its atomic subsets $H_0,\ldots,H_n$ such the singletons among $H_0,\ldots,H_n$ are $H_0,\ldots H_{m-1}$. H_0,\ldots,H_n$. Choose a positive$\mu_A(H_i)\in R$for each$i\geq m$, i$, such that $\sum_{i=m}^n\mu_A(H_i)=\lambda_A(I)-m\delta$. For all atomic $K\in A\cap\mathrm{dom}(\lambda_A)$, define $\mu_A(K)=\lambda_A(K)$. \sum_{i\leq n}\mu_A(H_i)=(p-q)\delta+\sum_{i\leq k}(b_i-a_i)$. Now extend$\mu_A$from the atoms to all of$A$.) Let$U$be a fine ultrafilter on$\mathcal{E}$. ("Fine" means that$\{B\in\mathcal{E}:A\subseteq B\}\in U$for all$A\in\mathcal{E}$.) The ultraproduct measure$\mu_U$is$R^U$-valued and has the two properties you seek. 3 minor typos Yes, by compactness. Let$R$denote your favorite hyperreal ordered field and let$\delta\in R$be a positive infinitesimal. Let$\mathcal{E}$denote the set of all (standard) finite Boolean subalgebras of$\mathcal{P}([0,1))$. For every$A\in\mathcal{E}$, let$\lambda_A(I)$be the (exact) length of$I$for all half-open intervals$I\in A$; for all open or closed intervals$I\in A$, respectively subtract or add$\delta$to the length of$I$to define$\lambda_A(I)$; let$\lambda_A(S)=\delta$for all singletons$S\in A$. Extend$\lambda_A$to a probability measure$\mu_A$on$A$. (Specifically, for each minimal interval$I\in A$that is not an atom of$A$, partition$I$into its atomic subsets$H_0,\ldots,H_n$such the singletons among$H_0,\ldots,H_n$are$H_0,\ldots H_{m-1}$. Choose a positive$\mu_A(H_i)\in R_1$R$ for each $i\geq m$, such that $\sum_{i=m}^n\mu_A(H_i)=\lambda_A(I)-m\delta$. For all atomic $K\in A\cap\mathrm{dom}(\lambda_A)$, define $\mu_A(K)=\lambda_A(K)$. Now extend $\mu_A$ from the atoms to all of $A$.)
Let $U$ be a fine ultrafilter on $\mathcal{E}$. ("Fine" means that $\{B\in\mathcal{E}:A\subseteq B\}\in U$ for all $A\in\mathcal{E}$). A\in\mathcal{E}$.) The ultraproduct measure$\mu_U$is$R^U$-valued and has the two properties you seek. 2 minor correction to$\mu_A$construction; added 42 characters in body Yes, by compactness. Let$R$denote your favorite hyperreal ordered field and let$\delta\in R$be a positive infinitesimal. Let$\mathcal{E}$denote the set of all (standard) finite Boolean subalgebras of$\mathcal{P}([0,1))$. For every$A\in\mathcal{E}$, let$\lambda_A(I)$be the (exact) length of$I$for all half-open intervals$I\in A$; for all open or closed intervals$I\in A$, respectively subtract or add$\delta$to the length of$I$to define$\lambda_A(I)$; let$\lambda_A(S)=\delta$for all singletons$S\in A$. Extend$\lambda_A$to a probability measure$\mu_A$on$A$. (Specifically, for each minimal half-open interval$I\in A$that is not an atom of$A$, partition$I$into its atomic subsets$H_0,\ldots,H_n$such the singletons among$H_0,\ldots,H_n$are$H_0,\ldots H_{m-1}$. Choose a positive$\mu_A(H_i)\in R_1$for each$i\geq m$, such that$\sum_{i=m}^n\mu_A(H_i)=\lambda_A(I)-m\delta$. For all atomic$K\in A\cap\mathrm{dom}(\lambda_A)$, define$\mu_A(K)=\lambda_A(K)$. Now extend$\mu_A$from the atoms to all of$A$.) Let$U$be a fine ultrafilter on$\mathcal{E}$. ("Fine" means that$\{B\in\mathcal{E}:A\subseteq B\}\in U$for all$A\in\mathcal{E}$). The ultraproduct measure$\mu_U$is$R^U\$-valued and has the two properties you seek.