# Thread: Is this function (Lebesgue) Integrable?

1. ## Is this function (Lebesgue) Integrable?

Let f be a Lebesgue integrable function that's equal to 0 for all x not in [a,b]

Let g be a function that is equal to 0 for all x not in a interval [-d,d] and suppose that g is C^

Is the function h(x) = f(x)*g(x) integrable?

2. Originally Posted by southprkfan1
Let f be a Lebesgue integrable function that's equal to 0 for all x not in [a,b]

Let g be a function that is equal to 0 for all x not in a interval [-d,d] and suppose that g is C^

Is the function h(x) = f(x)*g(x) integrable?
Your function $g$ is bounded (continuous on a segment), hence $fg$ is integrable.

3. Originally Posted by Laurent
Your function $g$ is bounded (continuous on a segment), hence $fg$ is integrable.
I can "see" how that would follow, but I'm not entirely sure the formal proof.

EDIT: Would it be that:

$\int gf < \int Kf < \infty$ where K is the upper bound of g?

4. Originally Posted by southprkfan1
I can "see" how that would follow, but I'm not entirely sure the formal proof.

EDIT: Would it be that:

$\int gf < \int Kf < \infty$ where K is the upper bound of g?
No, it would be $\int |gf|\leq \int K|f|=K\int |f|<\infty$