A field theory is developed based on the idea that the effective action of a yet unknown fundamental theory, at an energy scale below the Planck mass Mp, has the form of expansion in two measures: 5=f d4x[ΦL1 + √-gL2], where the new measure Φ is defined using the antisymmetric tensor field Φd4x=∂[αA βγδ]dxα ΛdxβΛdxγΛdx δ. A shift L1→L1 + const does not affect the equations of motion, whereas a similar shift when implementing with L2 causes a change which in standard OR would be equivalent to that of the cosmological constant (CC) term. The next basic conjecture is that the Lagrangian densities L1 and L2 do not depend on Aμvλ. The new measure degrees of freedom result in the scalar field ξ=Φ/√/√-g alone. A constraint appears that determines ξ in terms of matter fields. After the conformal transformation to the new variables (Einstein frame), all equations of motion take the canonical OR form of the equations for gravity and matter fields and, therefore, the models we study are free of the well-known defects that distinguish the Brans-Dicke type theories from Einstein's GR. All novelty is revealed only in an unusual structure of the effective potentials and interactions which turn over our intuitive ideas based on our experience in field theory. For example, the greater A we admit in L2, the smaller magnitude of the effective infiaton potential U(φ) will there be in the Einstein picture. Field theory models are suggested with explicitly broken global continuous symmetry, which in the Einstein frame has the form φ→φ+const. The symmetry restoration occurs as φ → ∞. A few models are presented where the effective potential U(φ) is produced with the following shape: for φ≤ - Mp, Uφ has the form typical for inflation model, e.g., U=λφ4 with λ ∼ 10-14; for φ≲-Mp, U(φ) has mainly the exponential form U∼e -aφ/Mp with variable a; = 14 for -Mp≳φ≲Mp, which gives the possibility for nucleosynthesis and large-scale structure formation; and a = 2 for φ∼Mp, which implies the quintessence era. There is no need for any fine-tuning to prevent the appearance of the CC term or any other terms that could violate the flatness of U(φ) at φ ≫ is obtained without fine-tuning as well. Quantized matter field models, including spontaneously broken gauge theories, can be incorporated without altering the results mentioned above. Direct coupling of fermions to the inflaton resembles Wetterich's model, but there is a possibility to avoid any observable effect at the late universe. SSB does not raise any problems with the CC in the late universe.