Proving a theorem on Lebesgue integral, can someone help?

Question

Let (X, M u) be a measure space, where X is a set, M is a sigma-algebra on X and u is a measure defined on M. Let f_n be a sequence of functions defined on X and with values on [0, oo] such that lim f_n = f. Suppose that lim Integral f_n du = Integral f du < oo (the integrals taken over X).

Show that, for every set E of M, Integral_E f_n du = Integral_E f du, where Integral_E means integral over E. Also, show that this conclusion may fail if we have lim Integral f_n du = Integral f du = oo

Thank you.

Answer 1

The given conditions imply that, for sufficiently large n, f_n is integrable over X and, therefore, over each measurable set E. So, we can assume, without loss of generality, that such conditions hold for every n. Also, f is integrable over every E of M and lim f_n = f on each subset of X.

Let E' be the complement of E, so that E' is in M. Then, for every natural n,

Int_E f_n du + Int_E' f_n du = Int f_n du

Since lim Int f_n du = Int f du = Int_E f du + Int_E' f du, it follows that

lim (Int_E f_n du + Int_E' f_n du) = Int_E f du + Int_E' f du (1).

Since f_n is a sequence in L+, it follows from Fatou's Lemma that:

Int_E lim inf f_n du = Int_E lim f_n du = Int_E f du <= lim inf Int_E f_n du. So, emphasizing the last inequality,

Int_E f du <= lim inf Int_E f_n du (2)

and, similarly (skipping the details),

Int_E' f du <= lim inf Int_E' f_n du (3)

The terms involved in (2) and (3) are finite, since f is integrable over E and over E'.

We know that if a_n and b_n are real sequences such that (a_n + b_n) --> a + b, a and b in R, and

a <= lim inf a_n and b <= lim inf b_n , then

a_n --> a and b_n --> b. (The proof of this result is given at the end of this post, anyway)

So, applying this result with a_n = Int_E f_n du , b_n = Int_E' f_n du, a = Int_E f du and b = Int_E' f du and considering (1), (2) and (3), it follows that

lim Int_E f_n du = Int_E f du, proving the theorem.


To see this conclusion may fail if lim Int f_n du = Int f du = oo, we can take X = (0, oo), M = Lebesgue sigma-algebra on X , u = Lebesgue measure and

f_n(x) = 1/(nx) if x is in (0, 1]
f_n(x) = 1 if x is in (1, oo) n=1,2,3.... ....

Then, f_n conveges to the function f given by

f(x) = 0, if x is in (0, 1] and f(x) = 1 if x is in (1, oo). Let E = (0, 1]. Then, recalling in this case the Lebesgue and Riemann integrals yields the same value, we have. for every n,

Int_E f_n du = Int (0 to 1)1/nx dx = 1/n ln(x) (0 to 1) = oo, Int_(1,oo) f_n du = 1 * oo = oo and, therefore, Int f_n du = oo. It follows that lim Int_E f_n du = oo and lim Int f_n du = oo.

On the other hand, Int_E f du = 0, Int_(0, 1) f du = oo and Int f du = oo

So, we have lim Int f_n du = Int f du = oo, but lim_E f_n du = oo > 0 = Int_E f du. This shows the condtion Int f du < oo cannot be dropped.



In this proof, we used the following result:

If a_n and b_n are real sequences such that (a_n + b_n) --> a + b, a and b in R, and

a <= lim inf a_n and b <= lim inf b_n , then

a_n --> a and b_n --> b

Proof:

For every n,
a_n = (a_n + b_n) - b_n.

By the properties of lim inf and lim sup of sequences, it follows from the assumptions that

lim sup a_n <= lim sup(a_n + b_n) + lim sup(-b_n) = a + b - lim inf b_n <= a + b - b = a. Hence,

lim sup a_n <= a <= lim inf a_n => lim a_n = a. It follows immediately that lim b_n = b

Answer 2

some authors enable easy purposes to take ? whilst defining the Lebesgue sums, and upload one greater term to the sum that money owed for it (the term ? x ?({f(x)=?})). yet this reasons technical issues so, so some distance as i be attentive to, maximum authors decide to limit the easy purposes approximating the integrand to the actual extensive form device: this avoids precisely the ought to introduce added hypothesis concerning to summability and/or positivity contained in the statements of the convergence theorems. I went by way of my books (this is been a whilst considering that I regarded at degree concept), and the only one i found that defines the essential with easy functionality to the prolonged reals is a e book by potential of Dym and McKean on Fourier sequence; the classic books on degree concept and Integration (Halmos, Taylor, Bartle, and so on.) all limit the easy purposes whilst defining the essential.