The entropy is defined as $$S:=-\int_{\Omega} p(x)\log p(x) $$ where $p(x)$ is a probability measure. It is naturally that $p(x)$ is constrained by $\int_\Omega p(x)=1$. If this is the sole constraint, $p(x)$ should be uniform distribution, optimize $S$ with Lagrange multiplier: $$J[p]:=-S-\lambda \left(\int_\Omega p(x) -1\right)$$ $$\frac{\delta J}{\delta p}=\log p(x) + 1 -\lambda=0$$ yields $$p(x)=\exp(\lambda-1)$$ which […]