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Let $A \subset \mathbb{R}^n$ be an oben bounded subset. Let $f \in L^1(A)$ be a BMO function, i.e. $$ \frac{1}{|A|}\int_{A} | f(x) - f_A| dx < \infty $$ where $|A|$ is the Lebesgue measure of the set |A| and $f_A \colon = \frac{1}{A} \int_A f(y) dy$.

I want to prove that given a subset $B \subset A$, it holds: $$ \frac{1}{|B|}\int_{B} | f(x) - f_B| dx \le \frac{1}{|A|}\int_{A} | f(x) - f_A| dx. $$

It should be trivial but I can't prove it.

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    Are you sure about the definition of BMO? – 2017-01-27

1 Answers 1

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Let $A=[-2,2]$ and $B = [-1,1]$. Take $f = \text{sign}(x)\chi_{[-1,1]}$. Then, $f_A=f_B=0$. $$\int_A \left\lvert f(x)-f_A\right\rvert = \int_B\left\lvert f(x)-f_B\right\rvert = 2$$ However, as $|A|>|B|$, the inequality you suggested does not hold.