Imagine for simplicity that the juggler at some instant repeats himself, i.e. that the juggler and balls (with masses $M$ and $2m$, respectively) are in the precise same kinematic state at times $t_1$ and $t_2$.
Consider man + 2 balls as the system, and bridge, etc., as the environment.
Let $p(t)$ be (the vertical component of) the total momentum of the system.
Newton's second law applied to the system yields:
$$\dot{p}(t) ~=~ F_n(t) - F_g, \tag{1}\label{eq:1}$$
where
$$F_g~=~(M+2m)g,\tag{2}\label{eq:2}$$
and where $F_n(t)$ is the normal force from the bridge, which may vary in time $t$ as the juggler does his routine.$^1$
Because of our simplifying assumption of repeating states, we have
$$0~=~p(t_2)-p(t_1)~=~ \int_{t_1}^{t_2} F_n(t) \, \mathrm{d}t - (t_2-t_1)F_g, \tag{3}\label{eq:3}$$
or
$$F_g ~=~ \frac{1}{t_2-t_1} \int_{t_1}^{t_2} F_n(t) \, \mathrm{d}t ~=~\langle F_n \rangle.\tag{4}\label{eq:4} $$
But if the average $\langle F_n \rangle$ is $F_g$, then clearly at at least one instance $t_3\in [t_1,t_2]$, one must have$^2$
$$F_n(t_3)\geq F_g.\tag{5}\label{eq:5}$$
In other words, the bridge collapses.
$^1$ The juggler is allowed to do whatever motion he thinks would benefit his case. Whether he wants to jump with both feet leaving the bridge, lower his center-of-mass, or fall down, is up to him. It seems physically reasonable to assume that the normal force $F_n(t)$ is a piecewise continuous function of time $t\in [t_1,t_2]$, with only finitely many discontinuity points. In that case the integral $\int_{t_1}^{t_2} F_n(t) \, \mathrm{d}t$ can be defined using the Riemann integral without involving the technically more complicated Lebesgue integral.
(Also note that the mean value theorem does not apply for discontinuous functions, and from a mathematical purist point of view, the mean value theorem is not needed, i.e., the crucial ineq. $\eqref{eq:5}$ may be established with considerations that are even more elementary.)
$^2$ Indirect proof of ineq. $\eqref{eq:5}$: Assume
$$\forall t\in [t_1,t_2]:~ F_n(t)~<~ F_g.\tag{6}\label{eq:6}$$
Then
$$\int_{t_1}^{t_2} F_n(t) \, \mathrm{d}t ~<~ (t_2-t_1)F_g,\tag{7}\label{eq:7}$$
if we assume piecewise continuity $t\mapsto F_n(t)$. But eq. $\eqref{eq:7}$ is inconsistent with eq. $\eqref{eq:3}$. QED.
