Nice and simple Proof
Lemma on Lebesgue integral
*Let $ u $ be a real function, non negative and integrable over $RR^n $ .
If $ int_(RR^n) u = 0 $ then $ u $ is null a.e. ( almost everywhere).
*Proof
Let’s consider :
$A_0 = (x: u(x) ne 0 )$ and $A_k = ( x: |u(x)| > 1/k ) $ with $ k >=1 $.
Then for each $ k $ we have :
$ |A_k|/k <= int_(A_k) u <= int_(RR^n) u =0 $ .
All $A_k $ have therefore measure equal to $0 $.
Then being $A_0 = uu_(k>=1) A_k $ , also $A_0 $ has measure equal to $0$ and consequently : $u(x) = 0 $ a.e.
*Let $ u $ be a real function, non negative and integrable over $RR^n $ .
If $ int_(RR^n) u = 0 $ then $ u $ is null a.e. ( almost everywhere).
*Proof
Let’s consider :
$A_0 = (x: u(x) ne 0 )$ and $A_k = ( x: |u(x)| > 1/k ) $ with $ k >=1 $.
Then for each $ k $ we have :
$ |A_k|/k <= int_(A_k) u <= int_(RR^n) u =0 $ .
All $A_k $ have therefore measure equal to $0 $.
Then being $A_0 = uu_(k>=1) A_k $ , also $A_0 $ has measure equal to $0$ and consequently : $u(x) = 0 $ a.e.
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