We analyze the sources of free electrons that produce the large dispersion measures, DM $approx 300-1600$ (in units cm$^{-3}$ pc), observed toward fast radio bursts (FRBs). Individual galaxies typically produce DM $sim 25-60$ cm$^{-3}$ pc from ionized gas in their disk, disk-halo interface, and circumgalactic medium. Toward an FRB source at redshift $z$, a homogeneous IGM containing a fraction $f_{rm IGM}$ of cosmological baryons will produce DM $= (935~{rm cm}^{-3}~{rm pc}) f_{rm IGM} , h_{70}^{-1} I(z)$, where $I(z) = (2/3 Omega_m)[ { Omega_m(1+z)^3 + Omega_{Lambda} }^{1/2} - 1 ]$. A structured IGM of photoionized Ly-alpha absorbers in the cosmic web produces similar dispersion, modeled from the observed distribution, $f_b(N,z)$, of H I (Lya-forest) absorbers in column density and redshift with ionization corrections and scaling relations from cosmological simulations. An analytic formula for DM($z$) applied to observed FRB dispersions suggests that $z_{rm FRB} approx 0.2-1.5$ for an IGM containing a significant baryon fraction, $f_{rm IGM} = 0.6pm0.1$. Future surveys of the statistical distribution, DM($z)$, of FRBs identified with specific galaxies and redshifts, can be used to calibrate the IGM baryon fraction and distribution of Ly-alpha absorbers. Fluctuations in DM at the level $pm10$ cm$^{-3}$ pc will arise from filaments and voids in the cosmic web.
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