[Peterdlax reference and answer for functional analysis exercises] Chapter 7th application of the results of Hilbert Space

1. Prove the theorem of Rado-Nikodym when the measurement is limited by $ \ Sigma $.

 

 

Proof: Set $ \ Mu, \ nu $ to all non-negative measurements limited by $ \ Sigma $, split $ \ Bex x = \ cup _ {I = 1} ^ \ infty X_ I = \ cup _ {j = 1} ^ \ infty y_j \ EEx $ to make $ $ \ Bex \ Mu (x_ I) <\ infty, \ quad \ nu (y_j) <\ infty. \ EEx $ write $ \ Bex x = \ cup _ {I, j = 1} ^ \ infty (x_ I \ cap y_j ), \ EEx $ \ Bex \ Mu (x_ I \ cap y_j) <\ infty, \ quad \ nu (x_ I \ cap y_j) <\ infty. \ EEx $ result of a limited number of measurements, $ \ Bex \ nu (E _ {IJ }) = \ int _ {e _ {IJ} g _ {IJ} \ RD \ Mu, \ quad \ forall \ E _ {IJ} \ subset X_ I \ cap y_j. \ EEx $ \ Bex \ nu (e) = \ int_e \ sum _ {I, j} g _ {IJ} \ RD \ Mu, \ quad \ forall \ e \ subset X. \ EEx $

 

 

2. Verify that $ C_0 ^ \ infty (d) $ the two inner products above are Inner Product Spaces.

 

 

Proof: $ \ Bex \ int_d \ sum | f_j | ^ 2 \ RD x = 0 \ LRA f_j = 0 \ (\ forall \ J) \ lra f = \ const \ LRA f \ equiv 0.\ EEx $

 

 

 

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