Let f ∈ L p ( R ) for some 1 ≤ p &lt; ∞. Given t ∈ R , we set f t ( x ) := f ( x − t ).Here I'm not sure if we can view { f t } as a sequence of functions, since they are not enumerated in a discrete sense. Is it true that when t → 0, we have f t → f pointwise a.e. and | | f t | | p → | | f | | p ?I'm also not sure if it's possible to apply results such as Dominated convergence theorem and Fatou's lemma when the functions are indexed like this, even if we were to set g t ( x ) := f ( x − 1 / t ) and let t tend to infinity.

Question

Let   f  ∈      L    p    (      R    ) for some   1  ≤  p  &amp;lt;  ∞. Given   t  ∈      R  , we set       f    t    (  x  )  :=  f  (  x  −  t  ).Here I&#039;m not sure if we can view   {      f    t    } as a sequence of functions, since they are not enumerated in a discrete sense. Is it true that when   t  →  0, we have       f    t    →  f pointwise a.e. and       |        |        f    t        |              |        p    →      |        |    f      |              |        p  ?I&#039;m also not sure if it&#039;s possible to apply results such as Dominated convergence theorem and Fatou&#039;s lemma when the functions are indexed like this, even if we were to set       g    t    (  x  )  :=  f  (  x  −  1      /    t  ) and let   t tend to infinity.

vrtuljakc6 · Accepted Answer

One can easily extend the basic theorems MCT, Fatou, DCT to this situation. For example,       f    t    →  f in       L    p   as   t  →  0 means that the function   ϕ  :      R    →      R   defined as   ϕ  (  t  )  :=  ‖      f    t    −  f      ‖    p   is such that       lim          t      →      0        ϕ  (  t  )  =  0. For such real functions, checking something about limits is the same as checking along every sequence. Meaning that                              lim                      t            →            0                          ϕ        (        t        )                            =        0            if and only if for every sequence   {      t    n        }          n      =      1              ∞      , with       lim          n      →      ∞            t    n    =  0, we have                              lim                      n            →            ∞                          ϕ        (                  t          n                )                            =        0.            So, this allows you to reduce to the case of sequences and thus use your usual sequential variant of DCT/MCT/Fatou.

Let f ∈ L p </msup> ( <mrow class="MJX-TeXAtom-ORD"> <mi

Answered question

Answer & Explanation

New Questions in Pre-Algebra