Let f in L^1(R^2) with respect to the Lebesgue measure m*m on R^2. Prove that if int int_(R^2) f(x,y)dxdy=0

Chelsea Lamb 2022-09-25 Answered
Let f L 1 ( R 2 ) with respect to the Lebesgue measure m × m on R 2 . Prove that if
R 2 f ( x , y ) d x d y = 0 ,
then there exits a square S a , b = { ( x , y ) a x a + 1 , b y b + 1 }, such that
S a , b f ( x , y ) d x d y = 0.
I tried to show that the integral
[ a , a + x ] × [ b , b + y ] f ( s , t ) d s d t
is absolutely continuous by Fubini's Theorem and Fundamental Theorem. And by the countable additivity of integration, I proved the integral on the whole plane is still A.C. However, I could not directly apply a theorem like the IVT for the single variable functions.
Is there any theorem for the two-dimensional case?
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Answers (1)

Zackary Galloway
Answered 2022-09-26 Author has 17 answers
Let g ( a , b ) = S a b f ( s , t ) d s d t. g is a continuous real valued function on R 2 . If it is never zero it is always positive or always negative. [ Because its range is connected in R ]. If it is always positive then R 2 f ( s , t ) d s d t > 0 because this integral is the sum of integrlas over S n m as n , m vary over all integers.
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