We study monotone convex functions
ψ
:
L
0
(
Ω
,
F
,
P
)
→
(
−
∞
,
∞
]
\psi :{L}^0(\Omega ,\mathcal {F} ,\mathbb {P})\to (-\infty ,\infty ]
and derive a dual representation as well as conditions that ensure the existence of a
σ
\sigma
-additive subgradient. The results are motivated by applications in economic agents’ choice theory.