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5

2

$begingroup$

In the following I denote by $B(x,r)$ the ball centered at $x$ of radius $r$, and by $|B(x,r)|$ the measure of this ball.
I am trying to solve the following exercise.

Question

Let $f geq 0, R geq 0$ and $B_R$ a ball of radius $R$ in $mathbb R^d$.
Show that for every $0 < r< R$ we have
$$
int_B_R f leq C int_B_R left( frac1B(x,r) int_B(x,r) f right) d x,
$$
where $C$ is a constant only depending on $d$.

Attempt

When $f$ is bounded, using the Lebesgue differentiation theorem
I can show that there exists $R'$ such that for all $r in (0, R')$
it holds
$$
int_B_R f leq
2 int_B_R left( frac1B(x,r) int_B(x,r) f right) d x.
$$
So it remains to prove the inequality when $r in (R', R)$ and also generalize it for unbounded functions $f$. I am not sure how to proceed
from this

real-analysis measure-theory lebesgue-integral

share|cite|improve this question

edited Apr 8 at 8:51

Little Bird

asked Apr 8 at 1:51

Little BirdLittle Bird

716

$endgroup$

add a comment | 

5

2

$begingroup$

In the following I denote by $B(x,r)$ the ball centered at $x$ of radius $r$, and by $|B(x,r)|$ the measure of this ball.
I am trying to solve the following exercise.

Question

Let $f geq 0, R geq 0$ and $B_R$ a ball of radius $R$ in $mathbb R^d$.
Show that for every $0 < r< R$ we have
$$
int_B_R f leq C int_B_R left( frac1B(x,r) int_B(x,r) f right) d x,
$$
where $C$ is a constant only depending on $d$.

Attempt

When $f$ is bounded, using the Lebesgue differentiation theorem
I can show that there exists $R'$ such that for all $r in (0, R')$
it holds
$$
int_B_R f leq
2 int_B_R left( frac1B(x,r) int_B(x,r) f right) d x.
$$
So it remains to prove the inequality when $r in (R', R)$ and also generalize it for unbounded functions $f$. I am not sure how to proceed
from this

real-analysis measure-theory lebesgue-integral

share|cite|improve this question

edited Apr 8 at 8:51

Little Bird

asked Apr 8 at 1:51

Little BirdLittle Bird

716

$endgroup$

add a comment | 

$begingroup$

In the following I denote by $B(x,r)$ the ball centered at $x$ of radius $r$, and by $|B(x,r)|$ the measure of this ball.
I am trying to solve the following exercise.

Question

Let $f geq 0, R geq 0$ and $B_R$ a ball of radius $R$ in $mathbb R^d$.
Show that for every $0 < r< R$ we have
$$
int_B_R f leq C int_B_R left( frac1B(x,r) int_B(x,r) f right) d x,
$$
where $C$ is a constant only depending on $d$.

Attempt

When $f$ is bounded, using the Lebesgue differentiation theorem
I can show that there exists $R'$ such that for all $r in (0, R')$
it holds
$$
int_B_R f leq
2 int_B_R left( frac1B(x,r) int_B(x,r) f right) d x.
$$
So it remains to prove the inequality when $r in (R', R)$ and also generalize it for unbounded functions $f$. I am not sure how to proceed
from this

real-analysis measure-theory lebesgue-integral

share|cite|improve this question

edited Apr 8 at 8:51

Little Bird

asked Apr 8 at 1:51

Little BirdLittle Bird

716

$endgroup$

share|cite|improve this question

edited Apr 8 at 8:51

Little Bird

asked Apr 8 at 1:51

Little BirdLittle Bird

716

share|cite|improve this question

edited Apr 8 at 8:51

Little Bird

asked Apr 8 at 1:51

Little BirdLittle Bird

716

asked Apr 8 at 1:51

Little BirdLittle Bird

716

asked Apr 8 at 1:51

Little BirdLittle Bird

716