The majority of answers already given have already provided some insight into the obvious methodological flaws, so I will not dwell on that here.  Instead, I'll provide a few practical options on how to treat this data given that it's already collected and despite the flawed question. There are a few ways to handle this.  You could consider marking responses that did not meet your definition of a "valid responses" by treating the entire question as missing and then following any number of practices for handling item-nonresponse such as those discussed [here][1].

You might also consider scaling each response so that the percentages add to 100.  Assuming each response is recorded as a percentage, this can be done by re-coding each original response $y_{{old}_{ij}}$ $(j=1,2,3,4)$ of the 4 sub-components to your question (i.e. artistic activity, government support, private pension, activities not related with arts) into a new response  $y_{{new}_{ij}}$ as follows:

\begin{eqnarray*}
y_{new_{ij}} & = & \begin{cases}
0 & ,\,\text{for}\,\sum_{j=1}^{4}y_{old_{ij}}=0\\
\frac{y_{old_{ij}}}{\sum_{j=1}^{4}y_{old_{ij}}} & ,\,\text{for}\,\sum_{j=1}^{4}y_{old_{ij}}>0\\
Missing & ,\,Otherwise
\end{cases}
\end{eqnarray*}
So for example, say you had a respondent $i$ who answered as follows:

     A. (i=1) Artistic activity:  10% 
     B. (i=2) Government support: 0% 
     C. (i=3) Private pension: 30% 
     D. (i=1) Activities not related with arts:  40%

Then you'd recode as follows:

\begin{eqnarray*}
y_{new_{i1}} & = & \frac{10}{10+0+30+40}=\frac{10}{80}=12.5\%\\
y_{new_{i2}} & = & \frac{0}{10+0+30+40}=\frac{0}{80}=00.0\%\\
y_{new_{i3}} & = & \frac{30}{10+0+30+40}=\frac{30}{80}=37.5\%\\
y_{new_{i4}} & = & \frac{40}{10+0+30+40}=\frac{40}{80}=50.0\%
\end{eqnarray*}

Note that all the new percentages now add to 100%.  Whatever you do, please be sure you make any transformations very clear when reporting your results.

  [1]: https://ssc.ca/en/case-study/handling-item-nonresponse-surveys