
    Terry HEBERT, Sr., Plaintiff-Appellant-Appellee v. Arlene RODRIGUEZ, individually and in her official capacity as a police officer; Defendant-Appellant Craig Melancon, Chief, in his official capacity as Chief of Police for Thibodaux Police Department; Thibodaux Police Department; Thibodaux City; Defendants-Appellees.
    No. 10-30489.
    United States Court of Appeals, Fifth Circuit.
    May 10, 2011.
    
      Glenn Charles Mcgovern, Esq., Metairie, LA, for Plaintiff-Appellant-Appellee.
    David Winston Ardoin, Esq., Thibodeaux, LA, for Defendant-Appellant.
    Before WIENER, BENAVIDES, and STEWART, Circuit Judges.
   PER CURIAM:

After carefully considering this appeal in light of the briefs, oral argument, and pertinent portions of the record, we affirm.

First, we affirm the district court’s denial of Rodriguez’s motion for judgment notwithstanding the verdict on this is Hebert’s false-arrest claim. Hebert’s primary evidence in support of his false-arrest claim was his own testimony at trial. Rodriguez would have us discredit this testimony in light of prior inconsistent statements. But she neglects that these pretrial statements go to Hebert’s credibility as a witness, not the legal sufficiency of the evidence. The determination of a witness’s credibility “‘lies at the core of a jury’s factfinding function’ ” and is not the province of judges. In other words, it is irrelevant whether we are persuaded of Hebert’s version of the events in this case — what matters is that the jury was persuaded. The evidence here is sufficient to support the jury’s finding.

We also affirm the jury’s damages award. The trial record contains legally sufficient evidence to support all of Hebert’s damages claims: lost business income, emotional distress, and punitive damages. We also find that the punitive-damages award was not excessive.

We further affirm the district court’s award of attorney’s fees pursuant to 42 U.S.C. § 1988 — the court’s award was properly calculated and carefully explained.

We further affirm the district court’s dismissal of Hebert’s municipal-liability claims, which he argues sound in both federal and state law. As to his municipal-liability claim under 42 U.S.C. § 1983, Hebert failed to present evidence to support one of the very few exceptions to the general rule that local governments are liable only for “their own illegal acts.” Thus, the district court was correct to dismiss his claim. As to municipal liability under Louisiana law, we hold that Hebert waived this claim by failing to properly present it to the district court and to this court on appeal.

We further affirm the district court’s dismissal of Hebert’s defamation claims. Hebert has failed to show — as he must to overcome the district court’s grant of judgment as a matter of law — “a conflict in substantial evidence on each essential element” of his defamation claims.

Finally, we affirm the two evidentiary rulings Hebert now challenges. He argues that the district court erred by excluding the testimony of his expert witness, but this issue is mooted by the court’s proper dismissal of his municipal-liability claims. Hebert also argues that the district court erred in admitting, on a motion in limine, evidence of Hebert’s twenty-six-year-old felony conviction. However, Hebert waived his objection to the admission of this evidence by first presenting it himself on direct examination.

In conclusion, we AFFIRM in all respects the jury’s verdict and the rulings of the district court. 
      
       Pursuant to 5th Circuit Local Rule 47.5, the court has determined that this opinion should not be published and is not precedent except under the limited circumstances set forth in 5th Circuit Local Rule 47.5.4.
     
      
      . We review de novo a district court’s denial of a motion for judgment notwithstanding the jury's verdict, applying the same standard as the district court. Chevron USA, Inc. v. Aker Maritime, Inc., 604 F.3d 888, 893 (5th Cir.2010) (citing E. Tex. Med. Ctr. Reg'l Healthcare Sys. v. Lexington Ins. Co., 575 F.3d 520, 525 (5th Cir.2009)). "In an action tried by jury, a motion for judgment as a matter of law is a challenge to the legal sufficiency of the evidence supporting the jury’s verdict.” McBeth v. Carpenter, 565 F.3d 171, 176 (5th Cir.2009) (citing Harrington v. Harris, 118 F.3d 359, 367 (5th Cir.1997)).
     
      
      . 9B Charles A. Wright & Arthur Miller, Federal Practice and Procedure § 2526 (3d ed.2008) (quoting Battle v. United Parcel Serv., 438 F.3d 856, 862 (8th Cir.2006)).
     
      
      . Leonard v. Dixie Well Serv. & Supply, Inc., 828 F.2d 291, 293-94 (5th Cir.1987).
     
      
      . We review this award for an abuse of discretion, El Paso Indep. Sch. Dist. v. Richard R., 591 F.3d 417, 425 n. 9 (5th Cir.2009), asking whether the district court's factual findings were clearly erroneous, id., and reviewing its legal conclusions de novo, Dearmore v. City of Garland, 519 F.3d 517, 520 (5th Cir.2008); Bailey v. Mississippi, 407 F.3d 684, 686 (5th Cir.2005).
     
      
      . See Connick v. Thompson,-U.S.-, 131 S.Ct. 1350, 1359-60, 179 L.Ed.2d 417 (2011) (liability where official policymakers fail to adequately train their employees about their duty to avoid violating citizens’ rights); City of St. Louis v. Praprotnik, 485 U.S. 112, 127, 108 S.Ct. 915, 99 L.Ed.2d 107 (1988) (liability under a ratification theory when "authorized policymakers approve a subordinate's decision and the basis for it”); Monell v. Dep’t of Soc. Servs. of City of New York, 436 U.S. 658, 691, 98 S.Ct. 2018, 56 L.Ed.2d 611 (1978) (liability where "action pursuant to official municipal policy” caused the plaintiff’s injury).
     
      
      . Pembaur v. Cincinnati, 475 U.S. 469, 479, 106 S.Ct. 1292, 89 L.Ed.2d 452 (1986) (citing Monell, 436 U.S. at 665-83, 98 S.Ct. 2018).
     
      
      . The district court repeatedly attempted to clarify the nature of Hebert’s municipal-liability claims before dismissing them, and Hebert failed to present the state-law theory it now urges.
     
      
      . The municipal-liability section of Hebert’s opening appeal brief fails to cite even one legal authority, let alone a specific Louisiana authority that creates vicarious municipal liability for police torts. Arguments not raised in an appellant’s opening brief are waived. Valle v. City of Houston, 613 F.3d 536, 544 (5th Cir.2010) (Tharling v. City of Port Lavaca, 329 F.3d 422, 430 (5th Cir.2003)).
     
      
      . Anthony v. Chevron USA, Inc., 284 F.3d 578, 583 (5th Cir.2002).
     
      
      . Ohler v. United States, 529 U.S. 753, 760, 120 S.Ct. 1851, 146 L.Ed.2d 826 (2000) ("A [party] who preemptively introduces evidence of a prior conviction on direct examination may not on appeal claim that the admission of such evidence was error.”).
     