grandes-ecoles 2017 Q1

grandes-ecoles · France · x-ens-maths2__mp Proof Direct Proof of an Inequality

Let $m$ be a measure. Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function that admits a variance relative to $m$. Show that $fm$ is integrable. As a consequence, the real $$\operatorname { Var } _ { m } ( f ) = \int f ( x ) ^ { 2 } m ( x ) d x - \left( \int f ( x ) m ( x ) d x \right) ^ { 2 }$$ is well defined. Show that $\operatorname { Var } _ { m } ( f ) \geqslant 0$.

Let $m$ be a measure. Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function that admits a variance relative to $m$. Show that $fm$ is integrable. As a consequence, the real
$$\operatorname { Var } _ { m } ( f ) = \int f ( x ) ^ { 2 } m ( x ) d x - \left( \int f ( x ) m ( x ) d x \right) ^ { 2 }$$
is well defined. Show that $\operatorname { Var } _ { m } ( f ) \geqslant 0$.

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