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

  1. Nov 10, 2014 #1
    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})##
    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!!!
     
  2. jcsd
  3. Nov 10, 2014 #2

    Fredrik

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  4. Nov 10, 2014 #3
    Thank you so much!
    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...:(
     
  5. Nov 10, 2014 #4

    Fredrik

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    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.
     
  6. Nov 11, 2014 #5
    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!
     
  7. Nov 12, 2014 #6

    WWGD

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    Don't you mean the set of measurable functions with ## \int_X f^pd\mu ## exists?
     
  8. Nov 12, 2014 #7
    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##!
     
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