Let $f \in \mathcal { C } _ { b } ^ { 0 } ( [ 0 , + \infty [ )$. The purpose of this question is to prove that $$\left( \int _ { 0 } ^ { + \infty } f ( x ) d x \text { converges } \right) \Rightarrow \left( \lim _ { t \rightarrow 0 ^ { + } } \mathcal { L } ( f ) ( t ) = \int _ { 0 } ^ { + \infty } f ( x ) d x \right)$$ We assume that $\int _ { 0 } ^ { + \infty } f ( x ) d x$ converges. (a) Prove that the function $$F : x \in \left[ 0 , + \infty \left[ \mapsto \int _ { x } ^ { + \infty } f ( t ) d t \right. \right.$$ is well-defined, continuous and bounded on $\left[ 0 , + \infty \right[$, of class $\mathcal { C } ^ { 1 }$ on $] 0 , + \infty [$ and satisfies $F ^ { \prime } = - f$. (b) Prove that $\lim _ { t \rightarrow 0 ^ { + } } \mathcal { L } ( f ) ( t ) = \int _ { 0 } ^ { + \infty } f ( x ) d x$ by integration by parts.
Let $f \in \mathcal { C } _ { b } ^ { 0 } ( [ 0 , + \infty [ )$. The purpose of this question is to prove that
$$\left( \int _ { 0 } ^ { + \infty } f ( x ) d x \text { converges } \right) \Rightarrow \left( \lim _ { t \rightarrow 0 ^ { + } } \mathcal { L } ( f ) ( t ) = \int _ { 0 } ^ { + \infty } f ( x ) d x \right)$$
We assume that $\int _ { 0 } ^ { + \infty } f ( x ) d x$ converges.
(a) Prove that the function
$$F : x \in \left[ 0 , + \infty \left[ \mapsto \int _ { x } ^ { + \infty } f ( t ) d t \right. \right.$$
is well-defined, continuous and bounded on $\left[ 0 , + \infty \right[$, of class $\mathcal { C } ^ { 1 }$ on $] 0 , + \infty [$ and satisfies $F ^ { \prime } = - f$.
(b) Prove that $\lim _ { t \rightarrow 0 ^ { + } } \mathcal { L } ( f ) ( t ) = \int _ { 0 } ^ { + \infty } f ( x ) d x$ by integration by parts.