# Constraints in path integral and the Lagrange multiplier

I was reading some references on the slave-particle approach to the Kondo problem and Anderson model. It is known that the slave-particle is introduced in the large Hubbard $$U$$ limit of the system so that the Hilbert space for the localized orbital (e.g., $$f$$ electron) on a single site only include: $$\{ | 0 \rangle, |\uparrow \rangle, |\downarrow\rangle \}$$.

The constraint $$n_f \le 1$$ then becomes $$b^\dagger b + \sum_{\sigma} s^\dagger_\sigma s_\sigma =1$$ (here $$b$$ is boson and $$s_\sigma$$ is fermion). In the path integral, people say the measure of the integration becomes:

$$\int D(s^\dagger,s)D(b^\dagger,b)\delta(b^\dagger b + \sum_{\sigma} s^\dagger_\sigma s_\sigma-1)$$

and a Lagrange multiplier is then introduced which replace the Dirac delta function above by: $$\int D\lambda \ e^{i\lambda(b^\dagger b + \sum_{\sigma} s^\dagger_\sigma s_\sigma-1)}.$$

So here I want to ask how does the constraint transformed into the Dirac delta function of the field (at each space-time) above?