We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$, and that only finitely many $p_i$ are strictly positive. We set $$g_0(x) = \begin{cases} \mathbb{P}(X > x) & \text{if } x \geqslant 0 \\ 0 & \text{if } x < 0 \end{cases}$$ For all $g \in \mathscr{F}$ (the set of positive bounded functions with support in a segment of $\mathbb{R}^+$), we denote by $Lg$ the unique solution of (E) bounded with support in $\mathbb{R}^+$. A sequence $\left(t_k\right)_{k \geqslant 0}$ satisfies property $(\mathscr{P})$ if $t_k \rightarrow +\infty$ and there exists a continuous bounded function $\mu : \mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that for every piecewise continuous $g \in \mathscr{F}$, $$Lg\left(t_k\right) \rightarrow \int_0^{+\infty} g(t)\mu(t)\,dt \quad \text{when} \quad k \rightarrow +\infty$$ Show, using question 14f, that for all $g \in \mathscr{F} \cap \mathscr{C}^1(\mathbb{R}, \mathbb{R}^+)$ and $\ell \geqslant 0$, $$\int_0^{+\infty} g(t)(\mu(t+\ell) - \mu(t))\,dt = 0$$
We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$, and that only finitely many $p_i$ are strictly positive. We set
$$g_0(x) = \begin{cases} \mathbb{P}(X > x) & \text{if } x \geqslant 0 \\ 0 & \text{if } x < 0 \end{cases}$$
For all $g \in \mathscr{F}$ (the set of positive bounded functions with support in a segment of $\mathbb{R}^+$), we denote by $Lg$ the unique solution of (E) bounded with support in $\mathbb{R}^+$. A sequence $\left(t_k\right)_{k \geqslant 0}$ satisfies property $(\mathscr{P})$ if $t_k \rightarrow +\infty$ and there exists a continuous bounded function $\mu : \mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that for every piecewise continuous $g \in \mathscr{F}$,
$$Lg\left(t_k\right) \rightarrow \int_0^{+\infty} g(t)\mu(t)\,dt \quad \text{when} \quad k \rightarrow +\infty$$
Show, using question 14f, that for all $g \in \mathscr{F} \cap \mathscr{C}^1(\mathbb{R}, \mathbb{R}^+)$ and $\ell \geqslant 0$,
$$\int_0^{+\infty} g(t)(\mu(t+\ell) - \mu(t))\,dt = 0$$