1
$\begingroup$

Forgive me, I am new to measure theory. I am trying to prove the Dominated Convergence Theorem by assuming Fatou's Lemma, here is what I have so far.

Fatou's Lemma:

Let $\{f_n\} $ be a sequence of nonnegative measurable functions s.t.

$f_n \rightarrow f$ a.e. x as $n \rightarrow \infty$,

then $\int f \leq $ lim$_{n \rightarrow \infty}$ inf $\int f_n$.

The DCT states:

Let $\{f_n\}$ be sequence of measurable functions s.t.

$f_n \rightarrow f$ a.e. x as $n \rightarrow \infty$,

if in addition we have

$\vert f(x) \vert \leq g(x)$ ; $g(x)$ integrable,

then $\int \vert f_n - f \vert \rightarrow 0$, as $n \rightarrow \infty$.

I know one way to prove this is you can define a set of elements bounded above by integer values so the functions are supported on a set of finite measure allowing the use of the bounded convergence theorem.

Now Fatou's Lemma takes into consideration the nonnegative functions, something I cannot assume with the DCT, but since the $f_n$ are all bounded above by an integrable function $g(x)$ could I rewrite $g(x)$ as its decomposition into $g^+ - g^-$?

My intuition says the result will "pop out" if I had non negativity? OR am I missing something else here?

$\endgroup$
2
  • 1
    $\begingroup$ The condition $|f| \leq g$ a.e. implies $g \geq 0$ a.e. $\endgroup$
    – William M.
    Dec 19, 2018 at 23:22
  • $\begingroup$ omg I cannot believe I missed that, thanks!! $\endgroup$
    – homosapien
    Dec 22, 2018 at 20:53

1 Answer 1

4
$\begingroup$

$\int [g-f]=\int \lim \inf [g-f_n] \leq \lim \inf \int [g-f_n]$ which gives $\int g -\int f \leq \int g -\lim \sup \int f_n$ so $\lim \sup \int f_n \leq \int f$. Now replace $f$ by $-f$ and $f_n$ by $-f_n$ to get $\lim \inf \int f_n \geq \int f$. Hence $\int f_n \to \int f$. To get $\int |f_n-f| \to 0$ you simply have to replace $f$ by $0$, $g$ by $2g$ and $f_n$ by $|f_n-f|$

$\endgroup$
3
  • $\begingroup$ ok I will be frank, I am a bit lost in your method. I used what you said a bit: Let $\{f_n\}$ be a sequence of measurable functions and suppose there is an integrable function $g$ such that $\vert f_n \vert \leq g$, then we also have $\vert f \vert \leq g$ thus we can define a new sequence of measurable functions say $h_n(x):= 2g(x) - \vert f_n(x) - f(x) \vert$, then it is clear that $h_n \rightarrow 2g$ and that $h_n$ is non-negative and measurable sequence then by invoking Fatou Lemma we have $\int 2g \leq \int 2g - \lim \inf \int \vert f_n - f \vert \leq \lim \sup \int \vert f_n - f \vert$ $\endgroup$
    – homosapien
    Jan 3, 2021 at 22:46
  • $\begingroup$ somewhere I know we use this math.stackexchange.com/questions/1553866/… $\endgroup$
    – homosapien
    Jan 3, 2021 at 22:50
  • $\begingroup$ sorry I just want to have every detail ironed out in full. $\endgroup$
    – homosapien
    Jan 3, 2021 at 22:51

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .