Renewal equation on the whole line

The following integral equation is considered:ϕ(x)=g(x)+∫−∞∞ϕ(x−t) dF(t),where g∈L1≡L1(−∞,∞), dF is a Borel probability measure, possessing nonzero absolute continuous component; at least one of numbers ν±<+∞ and ν≠0, whereν+=∫0∞x dF(x), ν−=−∫−∞0x dF(x), ν=ν+−ν−, −∞⩽ν⩽+∞.It is proved, that this equation has a solution ϕ0=ϕ1+ϕ2, where ϕ1∈L1, ϕ2∈C(−∞,∞), ∃ finite limits ϕ2(±∞),ϕ2(+∞)−ϕ2(−∞)=ν−1∫−∞∞ g(x) dx. If ν=±∞, then ν−1=0.If g∈L1 is a bounded function and g(±∞)=0, then ϕ1 is bounded and ϕ1(±∞)=0.