QIII.A.2
Continuous Probability Distributions and Random Variables
Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions
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Let $f : C(0,1) \rightarrow \mathbb{R}$ be a continuous application. We define: $$\mathrm{N}_f(x,y) = \frac{1}{2\pi} \int_0^{2\pi} \mathrm{N}(x,y,t) f(\cos t, \sin t)\, \mathrm{d}t$$ on $D(0,1)$.
In this question, we fix $t_0 \in [0,2\pi]$, $(x,y) \in D(0,1)$ and $\varepsilon > 0$. Moreover, we denote, for all real $\delta > 0$: $$I_0^\delta = \left\{ t \in [0,2\pi] \mid \|(\cos t, \sin t) - (\cos t_0, \sin t_0)\|_2 \leqslant \delta \right\}$$
a) Show that $I_0^\delta$ is an interval or the union of two disjoint intervals.
The use of a drawing will be appreciated; however, this drawing will not constitute a proof.
b) Show, using the application $f$, the existence of a real $\delta > 0$ such that $$\left| \int_{t \in I_0^\delta} \mathrm{N}(x,y,t) \left( f(\cos t, \sin t) - f(\cos t_0, \sin t_0) \right) \mathrm{d}t \right| \leqslant \frac{\varepsilon}{2}$$
c) Let $\delta > 0$ be arbitrary. Show that, if $t \in [0,2\pi] \backslash I_0^\delta$ and $\|(x,y) - (\cos t_0, \sin t_0)\|_2 \leqslant \delta/2$, then $$|\mathrm{N}(x,y,t)| \leqslant 4 \frac{1 - (x^2 + y^2)}{\delta^2}$$
d) Deduce from the previous question that, for $\delta > 0$ fixed, there exists $\eta > 0$ such that, if $\|(x,y) - (\cos t_0, \sin t_0)\|_2 \leqslant \eta$, then $$\left| \int_{t \in [0,2\pi] \backslash I_0^\delta} \mathrm{N}(x,y,t) \left( f(\cos t, \sin t) - f(\cos t_0, \sin t_0) \right) \mathrm{d}t \right| \leqslant \frac{\varepsilon}{2}$$