Lebesgue Integral Solved Problems

Problem 1: Simple Example of a Lebesgue Integral

Problem: Compute the Lebesgue integral of the function
f(x) = 2 over the interval [0, 3] using the Lebesgue measure.

Solution:

First, recall that in the Lebesgue integral, we’re summing over the function values multiplied by the measure of the set where the function takes those values.
Since f(x) = 2 is constant for all x ∈ [0, 3], we can simplify the calculation.
We need to compute:
```
∫₀³ 2 dμ(x)
```
where μ(x) is the Lebesgue measure, and it assigns the “length” of the interval to the set.
For a constant function, this is simply the value of the function multiplied by the length of the interval:
```
∫₀³ 2 dx = 2 × (3 - 0) = 6
```

Conclusion: The Lebesgue integral of f(x) = 2 over [0, 3] is 6.

Problem 2: Lebesgue Integral of a Highly Oscillatory Function

Problem: Compute the Lebesgue integral of the oscillatory function
f(x) = sin(1/x) over the interval (0, 1].

Solution:

The function f(x) = sin(1/x) oscillates wildly as x → 0. To manage this, we’ll need to carefully apply the Lebesgue integral by breaking the function into smaller pieces.
To compute the Lebesgue integral, we will use the absolute integrability of the function and analyze whether this integral converges.
We need to compute:
```
∫₀¹ sin(1/x) dx
```
The oscillations in sin(1/x) are extreme as x → 0, but due to the bounded nature of the sine function (i.e., |sin(⋅)| ≤ 1), we can assess whether the integral of this oscillatory function converges or not.

Strategy:

Although the function oscillates as x → 0, the crucial idea is whether the oscillations “cancel out” enough to give a convergent integral. In fact, due to the high-frequency oscillations near 0, the integral doesn’t have a limit in the classical sense.

But we can study the absolute value of the function to determine whether it’s integrable in the Lebesgue sense.

Step-by-Step:

We examine the behavior of the integral of the absolute value:
```
∫₀¹ |sin(1/x)| dx
```
Since |sin(⋅)| ≤ 1, we know that:
```
∫₀¹ |sin(1/x)| dx ≤ ∫₀¹ 1 dx = 1
```
This shows that the absolute value of the oscillatory function is bounded and integrable over (0, 1].

Conclusion:

While we cannot easily compute the exact value of this integral (as it doesn’t have a simple expression), the key insight is that Lebesgue integration allows us to conclude that the oscillatory function f(x) = sin(1/x) is Lebesgue integrable over (0, 1], thanks to the boundedness of the sine function and the convergence of the integral.

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