I was working on an upperbound on a quantity that comes up in this problem: I am choosing, from a vector of $n$ elements only $k$ of them. When I considered $X$ as a discrete domain, the corresponding r.vs. $\mathcal{X}^n, \mathcal{X}^k$ were discrete random-vectors and the summation was something like $$\sum_{\mathcal{X}^k}\sup_{X^n}P(X^k=x^k|X^n)\leq \sum_{X^k} 1 = \binom nk,$$ (where for $X^k$ I mean a vector of $k$ elements $(x_1,\ldots,x_k): x_i\in X \quad \forall i$), as it represents the summation over all the possible vectors of size $k$ picked from a vector of size $n$, ignoring the probability value that this specific vector might have. Now I am trying to generalize this to integrals. The corresponding integral, I believe, is the following:
$$ \int_{X^k} \sup_{X^n}f_{X^k|X^n}(X^k|X^n) dX^k.$$
I am not even sure if it is correctly written. Can you help me to only understand how to correctly write the integral? Maybe some pinpoints on how to evaluate it? The upperbound of $1$ on the supremum shouldn't work with the pdf, is there a way to achieve a similar bound? I also thought of writing this as an expectation:
$$E_{X^k}\bigg[ \sup_{X^n}\bigg(\frac{f_{X^k|X^n}(X^k|X^n)}{f(X^k)}\bigg)\bigg],$$
where $f$ is the marginal pdf of $X^k$. But I don't know if this helps...