let f, g be L 1 ( R ) functions with Lebesgue measure. Define f t ( x ) = f ( x / t ) t . Prove f t ∗ g converges to a g in L 1 when t → 0 + , where a = ∫ R f ( x ) d x.my approach in brief: since f , g ∈ L 1 , by Tonelli-Fubini's theorem, we can show ∫ R f t ∗ g d x = ( ∫ R f ( x ) d x ) ∫ R g ( y ) d y ∫ R f t ∗ g d x = ( ∫ R f ( x ) d x ) ∫ R g ( x ) d x ∫ R ( f t ∗ g − ( ∫ R f ( x ) d x ) ∫ R g ( x ) ) d x = 0Therefore, ∫ R | f t ∗ g ( x ) − a g ( x ) | d x = 0I am feeling something is wrong with my approach. Do correct me and give me some hints for solving this simple question.

Question

let   f,  g be       L    1    (      R    ) functions with Lebesgue measure. Define       f    t    (  x  )  =            f      (      x              /            t      )        t  . Prove       f    t    ∗  g converges to   a  g in       L    1   when   t  →      0    +  , where   a  =      ∫                  R              f  (  x  )  d  x.my approach in brief: since   f  ,  g  ∈      L    1  , by Tonelli-Fubini&#039;s theorem, we can show      ∫                  R                  f    t    ∗  g  d  x  =      (          ∫                        R                      f    (    x    )    d    x    )        ∫                  R              g  (  y  )  d  y      ∫                  R                  f    t    ∗  g  d  x  =      (          ∫                        R                      f    (    x    )    d    x    )        ∫                  R              g  (  x  )  d  x      ∫                  R                  (          f      t        ∗    g    −          (              ∫                              R                              f      (      x      )      d      x      )              ∫                        R                      g    (    x    )    )    d  x  =  0Therefore,      ∫                  R                  |        f    t    ∗  g  (  x  )  −  a  g  (  x  )      |    d  x  =  0I am feeling something is wrong with my approach. Do correct me and give me some hints for solving this simple question.

Paxton James · Accepted Answer

We have  q      (    t    )    :=      ∫                  R                          |                  f                    t              ∗    g          (      x      )        −    a    g          (      x      )        |    d  x  ⩽      ∫                  R                          |                  f                    t                    (      y      )        |        (          ∫                        R                                        |      g              (        x        −        y        )            −      g              (        x        )            |        d    x    )    d  y  :=      ∫                  R                          |                  f                    t                    (      y      )        |        φ        (    y    )    d  ySuppose that   g is the indicator function of an interval       [          α        ,          β        ]  , then       φ        (    y    )    =  2     min      (          |      y      |        ,          β        −          α        )  , hence  q      (    t    )    ⩽  2      ∫                  R                  |    f          (      s      )        |    min      (    t          |      s      |        ,          β        −          α        )    d  swhich converges to 0 by the monotone convergence theorem when   t  →            0              +      .The general result follows by density of the space of step functions in             L              1            (          R        )  .

April Bush · Accepted Answer

The last inequality is a clever step.

let f , g be L 1 </msup> ( <mrow class="MJX-TeXAtom-ORD"> <mi

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