Re: The example of outer regular but not inner regular measure!

On 4월14일, 오후9시51분, David C. Ullrich <ullr...@xxxxxxxxxxxxxxxx> wrote:
On 13 Apr 2007 22:46:29 -0700, "Gary Cooper" <ghkrh...@xxxxxxxxxxx>
wrote:





Hi!

Some problem in Rudins's RCA shows that there is such a measure.

Define the distance between points X_1, X_2 in R^2 is given by
d(X_1, X_2) =|y_1-y_2| if x_1 = x_2 , d(X_1, X_2) =|y_1-y_2|+1 if x_1 !
= x_2 .
It is trivial to show that it is indeed a metric and the resulting
space is locally compact,
If f is the continuous function with compact support, then there is
only finite
x_1, x_2, ... x_(n-1), x_n such that for each x_i, f(x_i , y) is not
zero for at least one y.

Define a linear functional as follows;
¥Ëf=¥Ò_(j=1 to n) (integral_(from -inf to inf) f(x_j , y)dy ).

Probably that last line doesn't look the same as when you
entered it, right? When posting to usenet you need to use
only standard ASCII characters. You should have written

sum_{j=1}^n int_{-infinity}^infinity f(x_j, y) dy.

From this linear functional, we get the associated Borel measure ¥ì by
Riesz representation theorem s.t. for open set V, ¥ì(V)=sup{¥Ëf: f is a
continous function with compact support in V and 0<= f(x) <=1 } and
arbitrary E in sigma-algebra, ¥ì(E)=inf( ¥ì(V) : for all open set V
containing E}.

Let E is the x-axis.
I showed for every compact set in E has measure zero for ¥ì. But I
didn't prove u(E) is infinite. If this resolved, then Rudin, as he
asserted, illustrates one example of Borel measure which is outer
regular but inner regular.

Please help me to show u(E) is infinite.

Use the definition above: You need to show that if V is
open and E is contained in V then mu(V) = infinity.
What would such a V look like?

************************

David C. Ullrich- 따온 텍스트 숨기기 -

- 따온 텍스트 보기 -

Oh sorry. Taking your advice sincerely, next time, I will use only
standard ASCII characters.

I consider the figure of V as the union of N_(ε_x)(x) as x over E and
ε_x < 1. (here, N_(ε_x) (x) = (x, (-ε_x, ε_x) ) ). To compute the
measure of u(V), I applied the Uryshon lemma to generate the
function of C_c(X) s.t. K<f<V.
(K<f<V means that f is continuous function with compact support
(i.e. C_c(X)) and this support is in V and f is 1 in K and 0<=f<=1 for
all X)

But since K is finite, I cannot take a sequence of compacts K_n s.t.
linear functional of f to approach to infinite. If it is possible to
take a infinite sequence of neighborhood consisting fixed V whose
radius has a nonzero infimun, this problem could be easily proved, nut
I think this is not easy.
(For example, the case every rational points {r_n} of R_1 has a
neighborhood 1/2^n for each of r_n)

Please help me.



.

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