# Analysis Prelim prep: Lebesgue integration

• laonious
In summary, the problem is that the integral is not integrable. However, by breaking it up into finite and improper integrals, we can show that it is still integrable.
laonious
Hi everyone,
I am studying past analysis prelim exams to take in the fall and have run into one which really has me stumped:

Let f be a real-valued Lebesgue integral function on [0,\infty).
Define
F(x)=\int_{0}^{\infty}f(t)\cos(xt)\,dt.
Show that F is defined on R and is continuous on R.
Show that \lim_{\rightarrow \infty}F(x)=0.

That F is defined is straightforward I think, the part I am struggling with is showing that it is continuous.

Going the route of a direct proof:
F(x)-F(y)\$=\int_{0}^{\infty}f(t)(\cos(xt)-\cos(yt))\,dt,
but it isn't clear to me how to show that this is enough to make the integral small.
Any ideas would be greatly appreciated, thanks!

$$|\cos(xt)-\cos(yt)| <= |t||x-y||\sin(ct)| <= |t||x-y|$$ where we use the mean value theorem.

Also, $$|cos(xt)-cos(yt)| <= 2$$ for all x,y and t.

Thus $$|\int^{\infty}_0f(t)(\cos(xt)-\cos(yt)) dt| \leq \int^{\infty}_0|f(t)||\cos(xt)-\cos(yt)| dt \leq \int^{r}_0|f(t)||t||x-y| dt + \int^{\infty}_r2|f(t)| dt$$

Now, pick an epsilon. Let r be a real number such that $$\int^{\infty}_r2|f(t)| dt \leq \frac{\epsilon}{2}$$. We have that

$$\int^{r}_0|f(t)||t||x-y| dt \leq |x-y|r\int^{r}_0|f(t)| dt$$, so let $$\delta = \frac{\epsilon}{2r\int^{r}_0|f(t)| dt}$$. Thus $$|F(x)-F(y)| \leq \epsilon$$.

Wow, that's clever breaking it up into a finite and an improper integral. I'd tried using the mean value theorem but wasn't sure how to show integrability. Thanks for your help!

laonious said:
Wow, that's clever breaking it up into a finite and an improper integral. I'd tried using the mean value theorem but wasn't sure how to show integrability. Thanks for your help!

No problem. Note though we are not transitioning into improper and finite riemann-integrals if that was what you were implying; the conditions of f does not imply riemann-integrability. Think of it as applying the indicator function wrt the area over which it is integrated in my "riemann integral notation", something I suppose the problem itself assumed by the wording of it.

(That such an r exists is not immediately obvious, we use that u(A) = int_A 2|f(t)| dt (lebesgue-integral) is a measure , and that lim_(A \to emptyset) u(A) = 0. It can perhaps be proven more easily.)

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## 1. What is Lebesgue integration and how is it different from Riemann integration?

Lebesgue integration is a mathematical concept used to calculate the area under a curve or the volume under a surface. It differs from Riemann integration in that it takes into account the entire domain of the function, rather than just the values of the function at specific points.

## 2. What is the significance of the Lebesgue measure in Lebesgue integration?

The Lebesgue measure is a concept used to determine the size or extent of a set of numbers in a given space. In Lebesgue integration, it is used to determine the measure or "size" of the domain of a function, which is crucial in calculating the integral.

## 3. How is the Lebesgue integral defined and how is it calculated?

The Lebesgue integral is defined as the supremum of the lower sums of a function over all possible partitions of the domain. It is calculated by taking the limit of these lower sums as the partition becomes finer and finer. This is known as the Riemann integral.

## 4. What are some applications of Lebesgue integration in the field of mathematics?

Lebesgue integration has various applications in mathematics, including probability theory, functional analysis, and differential equations. It is also used in physics and engineering to model and solve problems involving continuous functions.

## 5. What are some common techniques used to solve problems involving Lebesgue integration?

Some common techniques for solving problems involving Lebesgue integration include using the Lebesgue Dominated Convergence Theorem, the Monotone Convergence Theorem, and Fubini's Theorem. Other techniques include using change of variables and integration by parts.

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