In a paper by Talagrand, the following theorem is proved.
Theorem 1.4. There exists a number $K$ with the following property. Consider $n$ independent random variables $X_i$ valued in a measurable space $\Omega$. Consider a (countable) class $\mathcal{F}$ of measurable functions on $\Omega$. Consider the random variable $Z =\sup_{f \in \mathcal{F}} \sum_{i \leq n} f(X_i)$. Consider $$ U = \sup_{f \in \mathcal{F}} \|f\|_\infty\quad \text{ and }\quad V = \mathbb{E}\left[\sup_{f \in \mathcal{F}} \sum_{i \leq n} f(X_i)^2\right] $$ Then for each $t > 0$, we have $$ P(\left|Z - \mathbb{E}[Z]\right| \geq t) \leq K \cdot \mathsf{exp}\left(-\frac{t}{KU}\log\left(1 + \frac{tU}{V}\right)\right) $$
My Question. What is the exact value of $K$? I need to use that inequality but since I am not familiar with measure theory, some derivations in the paper are hard for me to understand and I cannot find the value of $K$ by myself.