Question 6. In Example 2.4.3, show that f(r) converges to 0 at each r E [0, 1].
. In Example 2.4.3, show that fn(x) converges to 0 at each x ∈ [0, 1].
Define fn by
fn(x) = n X(0, 1
n ](x) = n if 0 < x ≤ 1
n ,
0 otherwise.
In this case limn→∞ fn(x) = 0 for all x ∈ [0, 1]. So here is an example
where the pointwise limit of a sequence of Lebesgue integrable functions is Lebesgue integrable. The odd thing is that 1
0
fn = 1 for
every n, but 1
0
0 = 0. In other words, here is an example where
limn→∞ b
a
fn =
b
a
limn→∞ fn
![Question 6. In Example 2.4.3, show that fn (2) converges to 0 at each r € [0, 1].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F01416e22-b897-4a9e-9b93-2d1738197d91%2F3da40b04-7b52-4902-84ae-728fcfb71ffa%2F1p1fll_processed.jpeg&w=3840&q=75)
![Example 2.4.3. Define f. by
n if 0<r<
fa(x) = n X(@2(x) = {
%3D
(0.
otherwise.
In this case lim fn(x) 0 for all r E (0,1]. So here is an example
where the pointwise limit of a sequence of Lebesgue integrable fune-
tions is Lebesgue integrable. The odd thing is that fn = 1 for
every n, but
0 = 0. In other words, here is an example where
lim
What we are seeking are conditions that allow us to interchange
two limit-type operations, namely, integration (Lebesgue integration
in this case) and the limit of a sequence of functions. Our goal is a
theorem that addresses this issue, the Lebesgue Dominated Conver-
gence Theorem. Before the statement and proof of this theorem, we
need two lemmas.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F01416e22-b897-4a9e-9b93-2d1738197d91%2F3da40b04-7b52-4902-84ae-728fcfb71ffa%2Fa37ozgq_processed.jpeg&w=3840&q=75)
Let us consider the function :
We have to prove that converges to 0 for each
By the definition of for ,
Therefore, . ----------> (1)
Here we will first fix one and will show that . Now through the arbitraryness of the we can conclude [ since (1) also holds ]
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