Let F be a non-Archimedean local field, and μ F a Haar measure on F. The space C c ∞ ( F ) of locally constant functions of compact support is spanned by characteristic functions of the sets a + p b , for a ∈ F , b ∈ Z and p the maximal prime ideal of O F .I'm trying to prove the following: let Φ 0 , Φ 1 be the characteristic functions of O F , a + p b , respectively. If ∫ F Φ 0 ( x ) d μ F ( x ) = c 0 &gt; 0 ,then ∫ F Φ 1 ( x ) d μ ( x ) = c 0 q − b where q is the cardinality of the finite field O F / p and | | x | | = q − v F ( x ) for all x ∈ F. I feel to prove this I must express O F in some way related to the sets a + p b , and use properties of the Haar measure to find the measure of a + p b . Can it be answered like this, or is there another way?

Question

Let   F be a non-Archimedean local field, and       μ    F   a Haar measure on   F. The space       C    c          ∞        (  F  ) of locally constant functions of compact support is spanned by characteristic functions of the sets   a  +            p        b  , for   a  ∈  F  ,  b  ∈      Z   and       p   the maximal prime ideal of             O        F  .I&#039;m trying to prove the following: let       Φ    0    ,      Φ    1   be the characteristic functions of             O        F    ,  a  +            p        b  , respectively. If      ∫    F        Φ    0    (  x  )    d      μ    F    (  x  )  =      c    0    &amp;gt;  0  ,then      ∫    F        Φ    1    (  x  )    d  μ  (  x  )  =      c    0        q          −      b      where   q is the cardinality of the finite field             O        F        /        p   and       |        |    x      |        |    =      q          −              v        F            (      x      )       for all   x  ∈  F. I feel to prove this I must express             O        F   in some way related to the sets   a  +            p        b  , and use properties of the Haar measure to find the measure of   a  +            p        b  . Can it be answered like this, or is there another way?

Mekjulleymg · Accepted Answer

Your idea is correct. Note that by translation invariance, we may assume   a  =  0. Now note that            O        F    =      ⋃          x      ∈                        O                F                    /                              p                b              x  +            p        b  where we implicity chose representatives of             O        F        /              p        b  .By applying translation invariance and additivity, we get that  μ  (            O        F    )  =      |              O        F        /              p        b        |    ⋅  μ  (            p        b    )  =      q    b    μ  (            p        b    )

Let F be a non-Archimedean local field, and μ F </msub> a

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