Let $m$ be a measure. We assume that there exists a constant $C > 0$ such that for $\lambda \geq 0$, $$\int e ^ { \lambda f ( x ) } m ( x ) d x \leqslant \exp \left( \lambda \int f ( x ) m ( x ) d x + \frac { C \lambda ^ { 2 } } { 4 } \right) \tag{3}$$ applies (in particular to $f(x) = x$). 13a. Let $M = \int x m ( x ) d x$ and $a \geqslant M$. Show that $$\int _ { a } ^ { + \infty } m ( x ) d x \leqslant \exp \left( - \frac { ( a - M ) ^ { 2 } } { C } \right)$$ 13b. Conclude that for all $\alpha < \frac { 1 } { C }$, the function $x \mapsto e ^ { \alpha x ^ { 2 } } m ( x )$ is integrable on $\mathbb { R }$.
Let $m$ be a measure. We assume that there exists a constant $C > 0$ such that for $\lambda \geq 0$,
$$\int e ^ { \lambda f ( x ) } m ( x ) d x \leqslant \exp \left( \lambda \int f ( x ) m ( x ) d x + \frac { C \lambda ^ { 2 } } { 4 } \right) \tag{3}$$
applies (in particular to $f(x) = x$).
13a. Let $M = \int x m ( x ) d x$ and $a \geqslant M$. Show that
$$\int _ { a } ^ { + \infty } m ( x ) d x \leqslant \exp \left( - \frac { ( a - M ) ^ { 2 } } { C } \right)$$
13b. Conclude that for all $\alpha < \frac { 1 } { C }$, the function $x \mapsto e ^ { \alpha x ^ { 2 } } m ( x )$ is integrable on $\mathbb { R }$.