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A certain functional $T$ is defined as: $$T(F)=\int_{[\alpha,1-\alpha]}F^{-1}(s)M(ds)$$ where $M$ is a probability measure,for $\alpha>0$. The result that above functional is continuous at $F$, if the measure $M$ does not assign mass to the point where $F^{-1}$ is discontinuous is special case of Theorem 3.7. Robust Statistics by Huber and Ronchetti(2nd ed).

If $\frak{M}$ be a set of probability measures, is there a way to guarantee that $$T_{sup}(F)=\sup_{\{M\in\frak{M}\}}\{\int_{(0,1)}F^{-1}(s)M(ds)\}$$ is continuous if none of $M$ assign mass to point where $F^{-1}$ is discontinuous? Is it false?

We can at least say that it is lower semi continuous. Can we say more?

Details Added:

Domain of $T$ is CDFs. $F^{-1}(x):=\inf\{y\in\mathbb{R}:F(y)\ge x\}$

Edit The integral should be $$T_{sup}(F)=\sup_{\{M\in\frak{M}\}}\{\int_{[\alpha,1-\alpha]}F^{-1}(s)M(ds)\}$$ for $\alpha>0$

A certain functional $T$ is defined as: $$T(F)=\int_{(0,1)}F^{-1}(s)M(ds)$$ where $M$ is a probability measure with support $[\alpha,1-\alpha]$,for $\alpha>0$. The result that above functional is continuous at $F$, if the measure $M$ does not assign mass to the point where $F^{-1}$ is discontinuous is special case of Theorem 3.7. Robust Statistics by Huber and Ronchetti(2nd ed).

If $\frak{M}$ be a set of probability measures, is there a way to guarantee that $$T_{sup}(F)=\sup_{\{M\in\frak{M}\}}\{\int_{(0,1)}F^{-1}(s)M(ds)\}$$ is continuous if none of $M$ assign mass to point where $F^{-1}$ is discontinuous? Is it false?

We can at least say that it is lower semi continuous. Can we say more?

Details Added:

Domain of $T$ is CDFs. $F^{-1}(x):=\inf\{y\in\mathbb{R}:F(y)\ge x\}$

Edit

The integral should be $$T_{sup}(F)=\sup_{\{M\in\frak{M}\}}\{\int_{[\alpha,1-\alpha]}F^{-1}(s)M(ds)\}$$ for $\alpha>0$

A certain functional $T$ is defined as: $$T(F)=\int_{(0,1)}F^{-1}(s)M(ds)$$ where $M$ is a probability measure. The result that above functional is continuous at $F$, if the measure $M$ does not assign mass to the point where $F^{-1}$ is discontinuous is special case of Theorem 3.7. Robust Statistics by Huber and Ronchetti(2nd ed).

If $\frak{M}$ be a set of probability measures, is there a way to guarantee that $$T_{sup}(F)=\sup_{\{M\in\frak{M}\}}\{\int_{(0,1)}F^{-1}(s)M(ds)\}$$ is continuous if none of $M$ assign mass to point where $F^{-1}$ is discontinuous? Is it false?

We can at least say that it is lower semi continuous. Can we say more?

Details Added:

Domain of $T$ is CDFs. $F^{-1}(x):=\inf\{y\in\mathbb{R}:F(y)\ge x\}$

A certain functional $T$ is defined as: $$T(F)=\int_{[\alpha,1-\alpha]}F^{-1}(s)M(ds)$$ where $M$ is a probability measure,for $\alpha>0$. The result that above functional is continuous at $F$, if the measure $M$ does not assign mass to the point where $F^{-1}$ is discontinuous is special case of Theorem 3.7. Robust Statistics by Huber and Ronchetti(2nd ed).

If $\frak{M}$ be a set of probability measures, is there a way to guarantee that $$T_{sup}(F)=\sup_{\{M\in\frak{M}\}}\{\int_{(0,1)}F^{-1}(s)M(ds)\}$$ is continuous if none of $M$ assign mass to point where $F^{-1}$ is discontinuous? Is it false?

We can at least say that it is lower semi continuous. Can we say more?

Details Added:

Domain of $T$ is CDFs. $F^{-1}(x):=\inf\{y\in\mathbb{R}:F(y)\ge x\}$

Edit The integral should be $$T_{sup}(F)=\sup_{\{M\in\frak{M}\}}\{\int_{[\alpha,1-\alpha]}F^{-1}(s)M(ds)\}$$ for $\alpha>0$

A certain functional $T$ is defined as: $$T(F)=\int_{(0,1)}F^{-1}(s)M(ds)$$ where $M$ is a probability measure. The result that above functional is continuous at $F$, if the measure $M$ does not assign mass to the point where $F^{-1}$ is discontinuous is special case of Theorem 3.7. Robust Statistics by Huber and Ronchetti(2nd ed).

If $\frak{M}$ be a set of probability measures, is there a way to guarantee that $$T_{sup}(F)=\sup_{\{M\in\frak{M}\}}\{\int_{(0,1)}F^{-1}(s)M(ds)\}$$ is continuous if none of $M$ assign mass to point where $F^{-1}$ is discontinuous? Is it false?

We can at least say that it is lower semi continuous. Can we say more?

Details Added:

Domain of $T$ is Random Variables. Here, we can consider domain of $T$ to be all bounded random variables. $F^{-1}(x):=\inf\{y\in\mathbb{R}:F(y)\ge x\}$

A certain functional $T$ is defined as: $$T(F)=\int_{(0,1)}F^{-1}(s)M(ds)$$ where $M$ is a probability measure. The result that above functional is continuous at $F$, if the measure $M$ does not assign mass to the point where $F^{-1}$ is discontinuous is special case of Theorem 3.7. Robust Statistics by Huber and Ronchetti(2nd ed).

If $\frak{M}$ be a set of probability measures, is there a way to guarantee that $$T_{sup}(F)=\sup_{\{M\in\frak{M}\}}\{\int_{(0,1)}F^{-1}(s)M(ds)\}$$ is continuous if none of $M$ assign mass to point where $F^{-1}$ is discontinuous? Is it false?

We can at least say that it is lower semi continuous. Can we say more?

Details Added:

Domain of $T$ is CDFs. $F^{-1}(x):=\inf\{y\in\mathbb{R}:F(y)\ge x\}$