The softmax $\log \sum_{i=1}^n \exp(f_i)$ of vector $f$ is a smooth upper bound on $\max_i f_i$. However, the same cannot be said of $\log \int_{X} \exp(f(x))dx$ in relation to $\max_{x \in X} f(x)$ (assuming its existence) for some $X \subseteq \mathbb{R}^m$.
My question is: Is there a softmax (or something close) for continuous functions?
If I define \begin{align} g_1&=\ln\left(\int\limits_X e^{f(x)}dx\right)\\ g_2&=\ln\ln\left(\int\limits_X e^{e^{f(x)}}dx\right)\\ g_n&=\underbrace{\ln\ln\cdots\ln}_{n}\left(\int\limits_X \underbrace{e^{e^{^{.^{.^{.^{}}}}}}}_{n}{}^{^{^{^{^{^{f(x)}}}}}}dx\right) \end{align} then I see that $g_2$ does the job better than $g_1$, and $g_3$ does better than $g_2$.
I think that $g_3$ is already good enough for most of the elementary functions, but obviously higher $n$ would work better.