Evaluate integral wrt Lebesgue measure and find the L^p space if it exists

Question

I have been given the following the question:

Consider the following function $f : \mathbb R → \mathbb R$,

$$f=2·1_{(-3,1]}-3·1_{[5,+\infty)}$$

Here $1_A$ denotes the indicator function of set A.

Evaluate the integral $∫_\mathbb R fdλ$, with respect to the Lebesgue measure λ, if it exists. Does function f belong to the $L^p (\mathbb R,λ)$ space for some $p ≥ 1$?

I have said that the integral is: $2·(1-(-3))-3·(+\infty -5)= - \infty$

Is this true?

And if so, what $L^p$ space does this belong to, any hints?

All help is greatly appreciated.

Answer 1

The function $f$ is measurable and the second thing to check is whether the integral of $\lvert f\rvert$ is finite. Since $$ \left\lvert f\right\rvert =2·1_{(-3,1]}+3·1_{[5,+\infty)}, $$ the integral of $\lvert f\rvert$ is infinite.

Moreover, $$ \left\lvert f\right\rvert^p =2^p·1_{(-3,1]}+3^p·1_{[5,+\infty)} $$ hence $f$ does not belong to any $\mathbb L^p$-space.