Can this exercise be solved by DCT, I was only able to use MCT.

Question

How would you solve this exercise? You don't need to give me the details, just the general idea.

Let f be a Lebesgue integrable function. Show that

$\int f(x+a) d\lambda=f(x) d\lambda$ and

$\int f(ax)d\lambda=1/|a|*\int f(x)d\lambda$.

Hint: Show that it holds for f = $X_A$, where $A \in M$.

I solved this using the MCT, but what is troubling is that this exercise is under the chapter for DCT. So I think I was supposed to use the DCT in some way.

Here is how I solved it:

1. I showed it for the hint.
2. If it holds for the hint, it also holds for linear combinations for the hint, hence it holds for simple functions.
3. I wrote f = $f^+-f^-$, and I used the fact that we can find a series of non-nogative simple functions that converges to the two on THE rhs.
4. Then I used the property proved for simple functions on each term in the series. And combined with the MCT this gives our result.

Is there another way I should have done this? Without using MCT but DCT?

Answer 1

For any measurable function $f$ there exists a sequence of simple functions $f_n$ such that $|f_n| \leq |f|$ and $f_n \to f$ almost everywhere. Consequently, if we know that the claim holds for simple functions, we find by the dominated convergence theorem e.g. that

$$\int f(x+a) \, d\lambda(x) = \lim_{n \to \infty} \int f_n(x+a) \, d\lambda(x) = \lim_{n \to \infty} \int f_n(x) \, d\lambda(x) = \int f(x) \, dx$$

for any $f \in L^1$.