noted that this spectral index is derived from non-simultaneous observations and can thus be affected by variability effects (unless changes in the radio emission occur on timescales longer than years). Therefore, deeper 5 GHz observations simultaneous with new 1.6 GHz ones are needed before we can determine the nature of the radio emission.

X-ray luminosity function
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### AGN fraction

In order to estimate the level of AGN contamination, we compared the flux distribution of the *Chandra* sources with the expected AGN flux distribution (@2012ApJ...752...46L). We estimate that about 2/3 of the thirty-one *Chandra* sources detected inside the full S3 chip (area = 0.0196 deg$^{2}$) are likely to be AGN. However, for the fifteen sources inside the D25 ellipse (area = 0.0028 deg$^{2}$), the AGN fraction is only $\sim$19% (i.e. $\leq$ 3 AGN). In particular, the probability of finding an AGN inside the D25 with the flux measured for the ULX is $\sim$15%. Although the probability of detecting a background AGN inside the D25 of NGC4088 is not very low, other considerations argue against this possibility for the ULX (see Section \[discussion\]).

### XLF fitting {#XLF}

The XLF of the X-ray source population of NGC4088 is constructed considering those sources 