p*****n
发帖数: 758 | 1 这种问题不知如何下手，请教。
let f be a stricyly positive borel measurable function on the reals R, and
let E be a borel measurable subset of R with strictly positive Lebesgue
measure. For every t, define g(t)=integral of f(t+x)dx over E, show that g(t
) is borel measurable in t. | Q***5
发帖数: 994 | 2 For simplicity, assume measure of E is finite -- by taking limit, the
conclusion can also be proved when measure of E is infinite.
Let f_n(x) = min(n, f(x)) and g_n(t) = \int_E f_n(t+x)dx, prove that g_n is
continuous (hence Borel measurable) and that g_n converges to g point-wise.
(t
【在 p*****n 的大作中提到】
: 这种问题不知如何下手，请教。
: let f be a stricyly positive borel measurable function on the reals R, and
: let E be a borel measurable subset of R with strictly positive Lebesgue
: measure. For every t, define g(t)=integral of f(t+x)dx over E, show that g(t
: ) is borel measurable in t.
| p*****n
发帖数: 758 | 3 i see. The thing is i dont know how to show the lebesgue measure of the
symmetric difference of E and E+t goes to zero as t goes to zero.
is
wise.
【在 Q***5 的大作中提到】
: For simplicity, assume measure of E is finite -- by taking limit, the
: conclusion can also be proved when measure of E is infinite.
: Let f_n(x) = min(n, f(x)) and g_n(t) = \int_E f_n(t+x)dx, prove that g_n is
: continuous (hence Borel measurable) and that g_n converges to g point-wise.
:
: (t
| Q***5
发帖数: 994 | 4 You can prove that for the case where E is a bounded open set, using
dominated convergence Th.
For more general E, you may find open set A, such that E is a subset of A,
but m(A\E) is as small as you want.
Now, E\E+t is a subset of A\E+t, which in turn is a subset of (A\A+t) U (A+
t\E+t), and continuity follows
【在 p*****n 的大作中提到】
: i see. The thing is i dont know how to show the lebesgue measure of the
: symmetric difference of E and E+t goes to zero as t goes to zero.
:
: is
: wise.
| a***s
发帖数: 616 | 5 If you are allowed to use Fubini, then it is really straightforward
f(t+x)*1_E(t+x)
is Borel measurable in R*R
这种问题不知如何下手，请教。
let f be a stricyly positive borel measurable function on the reals R, and
let E be a borel measurable subset of R with strictly positive Lebesgue
measure. For every t, define g(t)=integral of f(t+x)dx over E, show that g(t
) is borel measurable in t.
【在 p*****n 的大作中提到】
: 这种问题不知如何下手，请教。
: let f be a stricyly positive borel measurable function on the reals R, and
: let E be a borel measurable subset of R with strictly positive Lebesgue
: measure. For every t, define g(t)=integral of f(t+x)dx over E, show that g(t
: ) is borel measurable in t.
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