In the context of learning theory, we usually have: data $(x,y)\sim P(x,y)$, with $x\in\mathcal{X}\subseteq\mathbb{R}^d$ and $y\in\mathcal{X}\subseteq\mathbb{R}^k$, a hypothesis class $\mathcal{F}\subseteq\Omega$, where $\Omega$ is the set of measurable functions $\mathcal{X}\rightarrow\mathcal{Y}$ and a loss function $\ell:\mathbb{R}^k\times\mathbb{R}^k\to\mathbb{R}_+$. Also, we define the functional $R$, the risk, such that $R(f)=\mathbb{E}_{xy}[\ell(y,f(x))]$
A Bayes model $f^*$ is defined as any function in $\Omega$ such that $\forall f\in\Omega$: $R(f^*)\leq R(f)$
A "best in class" $f^*_{F}$ is defined as any function in $\mathcal{F}$ such that $\forall f\in\mathcal{F}$: $R(f^*_F)\leq R(f)$
Given the random variable $D\sim P(x,y)^n$, define $R_{emp}(f; D)=\frac{1}{N}\sum_{i=1}^N\ell(y_i,f(x_i))$. Now, an empirical risk minimiser $f_{erm}$ is defined as any function in $\mathcal{F}$ such that $\forall f\in\mathcal{F}$: $R_{emp}(f_{erm})\leq R_{emp}(f)$.
My question is: what are the conditions needed to guarantee the existence of these functions? I'm not interested in the uniqueness. If one of these conditions is the continuity of $\ell$, do these quantities exist also for the $0/1$ loss?
