Let μ be a complex measure on R n and f , g ∈ L 1 ( μ ) such that 0 &lt; f ( x ) ≤ g ( x ) for a.e. x ∈ R n . Then is it true that | ∫ R n f d μ | ≤ | ∫ R n g d μ | ? I was trying to break each integral first in its real and imaginary part and then their corresponding positive and negative parts, i.e. ∫ R n f d μ = R e ( ∫ R n f d μ ) + i I m ( ∫ R n f d μ ) = R e + ( ∫ R n f d μ ) − R e − ( ∫ R n f d μ ) + i I m + ( ∫ R n f d μ ) − i I m − ( ∫ R n f d μ ) and then wanted to look at corresponding decompositions of the measure and the integral ∫ R n g d μ and obtain inequalities. But I can't figure them out and proceed from here.Can someone please help?

Question

Let   μ be a complex measure on             R              n       and   f  ,  g  ∈      L    1    (  μ  ) such that   0  &amp;lt;  f  (  x  )  ≤  g  (  x  ) for a.e.   x  ∈            R        n  . Then is it true that            |            ∫                            R                n              f  d  μ            |        ≤            |            ∫                            R                n              g  d  μ            |         ? I was trying to break each integral first in its real and imaginary part and then their corresponding positive and negative parts, i.e.      ∫                            R                n              f  d  μ  =  R  e            (            ∫                            R                n              f  d  μ            )        +  i  I  m            (            ∫                            R                n              f  d  μ            )        =  R      e    +              (            ∫                            R                n              f  d  μ            )        −  R      e    −              (            ∫                            R                n              f  d  μ            )        +  i  I      m    +              (            ∫                            R                n              f  d  μ            )        −  i  I      m    −              (            ∫                            R                n              f  d  μ            )      and then wanted to look at corresponding decompositions of the measure and the integral       ∫                            R                n              g  d  μ and obtain inequalities. But I can&#039;t figure them out and proceed from here.Can someone please help?

Fahrleine9m · Accepted Answer

If   μ  =      δ    0    −      δ    1   then the inequality becomes       |    f  (  0  )  −  f  (  1  )      |    ≤      |    g  (  0  )  −  g  (  1  )      |  . Can you come with an example where   0  &amp;lt;  f  ≤  g but this inequality fails? [ Make RHS 0].[Notation:       δ    x    (  E  )  =  1 if   x  ∈  E and 0 otherwise].

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