Re: cl(bigcap_{1<=q<=infty} L^q) = L^p

In article <6hjle.8288$uR4.7524@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Kira Yamato <no@xxxxxxxx> wrote:
>Ok, I'm trying to verify my often flawed logic, in this case, about
>density of L^p(R) spaces.
>
>Let f in L^p(R). Define the n-boxed function of f,
> f_n(x) = 0 if |x|>n,
> = n if f(x)>n,
> = -n if f(x)<n,
> = f(x) otherwise.
>Now, {f_n}, being bounded in domain and range, are clearly in L^q(R) for
>all 1<=q<=infty.
>
>Since clearly
> |f-f_n| <= |f|,
>Lebesgue's Dominated Convergence Theorem gives
> lim_n \int |f-f_n| = \int lim_n |f-f_n| = 0.
>So,
> f_n -> f strongly in L^p.
>
>But {f_n} are also in every other L^q spaces. So, we conclude that
> bigcap_{1<=q<=infty} L^q
>is dense in L^p.
>
>Is this correct or should I retake Real Variable I? Thanks ahead for
>helping me.

The conclusion in the text of your message is correct. However, the
equality in the title of your post is not. Certainly, L^p is in the
closure you specify, but it is not all of that closure. Therefore,
the equality is not warranted.

Rob Johnson <rob@xxxxxxxxxxxxxx>
take out the trash before replying
.

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