Approximation of ##f\in L_p## with simple function ##f_n\in L_p##

[DavideGenoa]

Dear friends, let us use the definition of Lebesgue integral on $X,\mu(X)<\infty$ as the limit

$\int_X fd\mu:=\lim_{n\to\infty}\int_Xf_nd\mu=\lim_{n\to\infty}\sum_{k=1}^\infty y_{n,k}\mu(A_{n,k})$
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where $\{f_n\}$ is a sequence of simple, i.e. taking countably many values $y_{n,k}$ for $k=1,2,\ldots$, functions $f_n:X\to\mathbb{C}$ uniformly converging to $f$, and $\{y_{n,k}\}=f_n(A_{n,k})$ where $\forall i\ne j\quad A_{n,i}\cap A_{n,j}=\emptyset$.I know that if any sequence $\{f_n\}\subset L_p(X,\mu)$, $p\geq 1$ uniformly converges to $f$ then [it also converges][1] with respect to the norm $\|\cdot\|_p$ to the same limit, which is therefore an element of $L_p(X,\mu)$.I read in Kolmogorov-Fomin's Elements of the Theory of Functions and Functional Analysis (1963 Graylock edition, p.85) that an arbitrary function $f\in L_2$ can be approximated [with respect to norm $\|\cdot\|_2$, I suspect] with arbitrary simple functions belonging to $L_2$.I do not understand how the last statement is deduced. If it were true, I would find the general case for $L_p$, $p\geq 1$, even more interesting. I understand that if $f\in L_p\subset L_1$ then for all $\varepsilon>0$ there exists a simple function $f_n\in L_1$ such that $\forall x\in A\quad |f(x)-f_n(x)|<\varepsilon$ and then, for all $p\geq 1$, $\|f-f_n\|_p<\varepsilon$, but how to find $f_n\in L_p$ (if $p=2$ or in general $p>1$)?Can anybody explain such a statement? Thank you so much!

 

[Fredrik]

I don't have a lot of time right now, but I think theorem 2.2.5 in Friedman is a good place to start. http://books.google.com/books?id=yT...an analysis&hl=sv&pg=PA34#v=onepage&q&f=false

 

[DavideGenoa]

Thank you so much!

Friedman is a good place to start

Very interesting book! As to the above said fact, I knew Friedman's theorem 2.2.5, but I am not able to prove what I said...:(

 

[Fredrik]

I'm not sure I understand. How you you define $L_p$? Is that not a set of measurable functions? (Did you mean $L^p$? For example, $L^2$ is the set of square-integrable functions.)

Friedman 2.2.5 tells you how to approximate an arbitrary measurable positive real-valued function by simple functions. The standard rewrite $f=f_+-f_-$ immediately generalizes this result to arbitrary measurable real-valued functions.

 

[DavideGenoa]

Oh, sorry, I intended the set of measurable functions such that $\int_X fd\mu$ exists by $L_p$, i.e. the same thing as $L^p$. Thank you again!

 

[WWGD]

Don't you mean the set of measurable functions with $\int_X f^pd\mu$ exists?

 

[DavideGenoa]

Don't you mean the set of measurable functions with $\int_X f^pd\mu$ exists?

Oh, yes, I mean by $f\in L_p(X,\mu)$ that $\int_X|f|^p d\mu$ exists. Excuse me: I omitted the $p$!