Correction to: Aequat. Math. 96 (2022), 833–841 https://doi.org/10.1007/s00010-022-00870-w

In the original article (published by P. Pasteczka) there is a mistake (observed by Árpád Száz) in the proof of Lemma 1. In fact, the statement is false in its original form, as it is demonstrated by the following counterexample: Let \( D = [ 0, + \infty [ \times [ 0, + \infty [ \) with the standard topology,

$$\begin{aligned} F (x_1,\, x_2)= & {} x_1 x_2 \qquad ((x_1,\, x_2) \in D), \\ T(s)= & {} [ 0, + \infty [ \times [0,s] \ \ \ (s \in [ 0, + \infty [ \, ). \end{aligned}$$

Then \( \textbf{F} \circ T (0) = \{ 0 \} \) and \( \textbf{F} \circ T (s) = [ 0, + \infty [ \) for every \( s \in ] 0, + \infty [ \,\).

However, in Theorem 1 it is sufficient to use the lemma for the restricted case when \(D=I^p\) (with the standard topology), where \(p \in \mathbb {N}\) and \(I \subset \mathbb {R}\) is a compact interval. The proper formulation should be

FormalPara Lemma 1

Let D be a compact, metrizable topological space and \(F :D \rightarrow [0,\infty )\) be a continuous function.

If \(T :[0,\infty ) \rightarrow 2^D\) is nondecreasing, right-continuous (we consider topological limit on \(2^D\)) and such that

  • each T(x) is closed;

  • \(\textbf{F} \circ T :[0,\infty ) \rightarrow 2^{[0,\infty )}\) is left-continuous.

Then \(\textbf{F} \circ T\) is constant.

Then the following modification is required in the proof:

Instead of:

However, as D is \(\sigma \)-compact, we obtain that \(D_0\) is compact.

There should be:

However \(D_0\) is compact being a closed subset of a compact set.

As the result of this improvement, the proof of Theorem 1 should start with:

Starting section of the proof: If there exist two different \(\textbf{M}\)-invariant means \(K_1,K_2 :I^p \rightarrow I\), then there exist a compact subset \(J \subseteq I\) and a vector \(v \in J^p\) such that \(K_1(v)\ne K_2(v)\). However \(K_i|_{J^p}\) is invariant with respect to \(\textbf{M}|_{J^p} :J^p \rightarrow J^p\) for \(i \in \{1,2\}\). Thus we can assume without loss of generality that I is a compact set.

Moreover in line 26, page 836, the following sentence should be added:

For the converse implication take any element \(y \in \textbf{F} \circ T (a)\). Obviously \(0 \in F \circ T(a^-)\), thus we can assume that \(y\ne 0\). Then, there exists...

More details related to this issue can be found in [5].