$\DeclareMathOperator\GL{GL}$Let $F$ be a $\mathfrak{p}$-adic field and $\mathscr{O}_{F}$ its valuation ring. For any measurable subset of $M_{d}(F)$ such as $$ A= \left( \begin{array}{ccc} a_{11}+t^{\alpha_{11}}\mathscr{O}_{F} & \cdots & a_{1d}+t^{\alpha_{1d}}\mathscr{O}_{F} \\ \vdots & \ddots & \vdots \\ a_{d1}+t^{\alpha_{d1}}\mathscr{O}_{F} & \cdots & a_{dd}+t^{\alpha_{dd}}\mathscr{O}_{F} \end{array} \right), $$ we define $\mu_{M_{d}(F)}(A)=(\frac{1}{q})^{\sum_{i, j}\alpha_{ij}}$, where $t$ is an uniformizer of $F$ and $q$ is the order of residue field of $F$. It is well-known that $\mu_{M_{d}(F)}$ is invariant for additive, that is, for any $a\in M_{d}(F)$, $\mu_{M_{d}(F)}(a+A)=\mu_{M_{d}(F)}(A)$.
I want to know a relation between $\mu_{M_{d}(F)}$ and a left (or right) Haar measure on $\GL_{d}(F)$. Let $\mu$ be the left Haar measure on $\GL_{d}(F)$ such that $\mu(\GL_{d}(\mathscr{O}))=1$. The measurable subset $A$ on $M_{d}(F)$ as above, when $A\subset \GL_{d}(F),$ what is the relationship between the value of $\mu(A)$ and $\mu_{M_{d}(F)}(A)$ ?
