Let $m$ be a measure. Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function that admits an entropy relative to $m$. We consider the function $h : [ 0 , + \infty [ \rightarrow \mathbb { R }$ defined by $h ( 0 ) = 0$ and for $x > 0$, $h ( x ) = x \ln ( x )$. 2a. Show that $f ^ { 2 } m$ is integrable. As a consequence, the real $$\operatorname { Ent } _ { m } ( f ) = \int h \left( f ( x ) ^ { 2 } \right) m ( x ) d x - h \left( \int f ( x ) ^ { 2 } m ( x ) d x \right)$$ is well defined. 2b. Let $a > 0$. Show that $$\forall x \geqslant 0 , \quad h ( x ) \geqslant ( x - a ) h ^ { \prime } ( a ) + h ( a ) ,$$ with strict inequality if $x \neq a$. 2c. Show that $\operatorname { Ent } _ { m } ( f ) \geqslant 0$. You may use the previous question with $a = \int f ( x ) ^ { 2 } m ( x ) d x$. 2d. We assume here that for all $x \in \mathbb { R } , m ( x ) > 0$. Characterize the functions $f$ such that $\operatorname { Ent } _ { m } ( f ) = 0$.
Let $m$ be a measure. Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function that admits an entropy relative to $m$. We consider the function $h : [ 0 , + \infty [ \rightarrow \mathbb { R }$ defined by $h ( 0 ) = 0$ and for $x > 0$, $h ( x ) = x \ln ( x )$.
2a. Show that $f ^ { 2 } m$ is integrable. As a consequence, the real
$$\operatorname { Ent } _ { m } ( f ) = \int h \left( f ( x ) ^ { 2 } \right) m ( x ) d x - h \left( \int f ( x ) ^ { 2 } m ( x ) d x \right)$$
is well defined.
2b. Let $a > 0$. Show that
$$\forall x \geqslant 0 , \quad h ( x ) \geqslant ( x - a ) h ^ { \prime } ( a ) + h ( a ) ,$$
with strict inequality if $x \neq a$.
2c. Show that $\operatorname { Ent } _ { m } ( f ) \geqslant 0$. You may use the previous question with $a = \int f ( x ) ^ { 2 } m ( x ) d x$.
2d. We assume here that for all $x \in \mathbb { R } , m ( x ) > 0$. Characterize the functions $f$ such that $\operatorname { Ent } _ { m } ( f ) = 0$.