#### Glasnik Matematicki, Vol. 36, No.1 (2001), 1-9.

### ON FUNCTIONS WHICH ARE ALMOST ADDITIVE MODULO A
SUBGROUP

### Janusz Brzdek

Department of Mathematics, Pedagogical Academy, Podchorazych,
30-084 Cracow, Poland

**Abstract.** Let (*X*,+) be a commutative semigroup,
uniquely divisible by 2, (*G*,+) be a topological group,
and *K* be a discrete, normal and contable subgroup of *G*.
We show that if *X* is endowed with a topology and the
topologies in *X* and *G* satisfy some additional
conditions, then for every measurable function *f* mapping
*X* into *G* such that *f*(*x*+*y*) -
*f*(*x*) - *f*(*y*)
∈ *K*
almost everywhere in *X*^{2}, with respect to some
ideal in *X*^{2}, there is an additive function
*A* : *X*
→ *G*
with *f*(*x*) - *A*(*x*)
∈ *K* almost
everywhere in *X*.

**1991 Mathematics Subject Classification.**
39B52.

**Key words and phrases.** Cauchy difference, measurability,
additive function, σ-ideal.

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